All practice questions

Question 2421Question →

In the standard $(x, y)$ coordinate plane, the graph of the parabola $y = x^2 - 5x + 2$ intersects the line $2x - y = 4$ at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the value of $x_1 x_2 + y_1 y_2$ ?

50

22

-10

6

102

Show answer & explanation

Answer: -10

Answer

The value of

x_1 x_2 + y_1 y_2

is

-10

.

The correct answer is

-10

. To find the intersection points, substitute the linear equation

y = 2x - 4

into the quadratic equation

y = x^2 - 5x + 2

, which simplifies to

x^2 - 7x + 6 = 0

. Factoring gives

(x-1)(x-6) = 0

, yielding

x

-coordinates

x_1 = 1

and

x_2 = 6

. Substituting these back into the linear equation gives the corresponding

y

-coordinates

y_1 = -2

and

y_2 = 8

. Evaluating the requested expression gives

(1)(6) + (-2)(8) = 6 - 16 = -10

.

Step-by-Step Solution

1

Isolate

y

in the linear equation.

y = 2x - 4

This allows for substitution into the quadratic equation.

2

Substitute

y = 2x - 4

into the quadratic equation

y = x^2 - 5x + 2

.

2x - 4 = x^2 - 5x + 2

Setting the two equations equal to each other helps find the x-coordinates of the intersection points.

3

Rearrange the equation into standard quadratic form.

x^2 - 7x + 6 = 0

This sets up the equation for factoring.

4

Factor the quadratic equation.

(x-1)(x-6) = 0

Factoring reveals the solutions for the x-coordinates.

5

Solve for the two x-coordinates.

x_1 = 1

and

x_2 = 6

These are the x-coordinates of the two intersection points.

6

Find the corresponding y-coordinates by substituting the x-values back into the linear equation.

For

x_1 = 1

:

y_1 = 2(1) - 4 = -2

. For

x_2 = 6

:

y_2 = 2(6) - 4 = 8

.

This gives the full coordinate pairs:

(1, -2)

and

(6, 8)

.

7

Calculate the value of the expression

x_1 x_2 + y_1 y_2

.

(1)(6) + (-2)(8) = 6 - 16 = -10

This provides the final required value.

Key Concept

Solving systems of linear and quadratic equations by substitution and evaluating expressions of their coordinate solutions.

Estimated Time:1m 30s

Question 2422Question →

What is the greatest integer value of $x$ that satisfies the inequality $3 - 2x > 10$ ?

Show answer & explanation

Answer: -4

Answer

The correct answer is

-4

.

Subtracting 3 from both sides of

3 - 2x > 10

gives

-2x > 7

. When dividing both sides by

-2

, the inequality sign must be flipped, yielding

x < -3.5

. The greatest integer less than

-3.5

is

-4

.

Step-by-Step Solution

1

Subtract 3 from both sides of the inequality to isolate the variable term.

-2x > 7

Subtracting 3 from both sides keeps the inequality balanced while moving the constant term to the right side.

2

Divide both sides by

-2

and reverse the inequality sign.

x < -3.5

Dividing or multiplying an inequality by a negative number requires reversing the direction of the inequality sign to maintain a true statement.

3

Identify the greatest integer that satisfies the inequality.

-4

The integers that are strictly less than

-3.5

are

-4, -5, -6, \dots

. The largest (greatest) of these integers is

-4

.

Key Concept

Solving linear inequalities by applying the sign-reversal rule when dividing by a negative number and identifying integer boundary values.

Question 2423Question →

In right triangle $DEF$ , the measure of $\angle E$ is $90^\circ$ and the measure of $\angle D$ is $60^\circ$ . If the hypotenuse $DF$ has a length of $14$ centimeters, what is the length, in centimeters, of the segment $DE$ ?

7\sqrt{2}

7\sqrt{3}

7

14\sqrt{3}

28

Show answer & explanation

Answer: $7$

Answer

The length of the segment

DE

is

7

centimeters.

The correct option is the one with the value

7

. In right triangle

DEF

, the angles are

90^\circ

,

60^\circ

, and

30^\circ

, making it a special

30^\circ

-

60^\circ

-

90^\circ

right triangle. The side

DE

is opposite the

30^\circ

angle (the shorter leg). By the properties of a

30^\circ

-

60^\circ

-

90^\circ

triangle, the shorter leg is half the length of the hypotenuse. Thus,

DE = 14 / 2 = 7

centimeters.

Step-by-Step Solution

1

Determine the measure of the third angle,

\angle F

.

\angle F = 30^\circ

The sum of angles in a triangle is

180^\circ

. Since

\angle E = 90^\circ

and

\angle D = 60^\circ

, we calculate

\angle F = 180^\circ - 90^\circ - 60^\circ = 30^\circ

.

2

Identify the relationship between the sides of the

30^\circ

-

60^\circ

-

90^\circ

triangle.

DE

is the shorter leg, opposite

\angle F

(

30^\circ

).

The side opposite the

30^\circ

angle is the shorter leg, which is half the length of the hypotenuse.

3

Calculate the length of

DE

.

DE = 7

centimeters

Since the hypotenuse

DF = 14

centimeters, the shorter leg

DE

is

14 / 2 = 7

centimeters.

Key Concept

In a

30^\circ

-

60^\circ

-

90^\circ

right triangle, the lengths of the sides are in the ratio

1 : \sqrt{3} : 2

. The shorter leg (opposite the

30^\circ

angle) is half the length of the hypotenuse.

Estimated Time:1m 0s

Question 2424Question →

In the standard $(x, y)$ coordinate plane, the circle defined by the equation $x^2 + y^2 - 12x - 4y + 31 = 0$ is translated $3$ units to the right and $5$ units down. A line passing through the origin with a non-zero slope $m$ is tangent to this translated circle. What is the value of $m$ ?

Show answer & explanation

Answer: -0.75

Answer

The correct answer is -0.75.

The correct answer is -0.75. By completing the square on the original equation, we find the circle

(x-6)^2 + (y-2)^2 = 9

with center

(6, 2)

and radius

3

. Translating the circle shifts the center to

(9, -3)

. A line passing through the origin with slope

m

has the equation

mx - y = 0

. For this line to be tangent to the circle, its perpendicular distance from the center

(9, -3)

must equal the radius

3

. Using the distance formula, we get the equation

\frac{|9m + 3|}{\sqrt{m^2 + 1}} = 3

. Simplifying gives

|3m + 1| = \sqrt{m^2 + 1}

. Squaring both sides yields

9m^2 + 6m + 1 = m^2 + 1

, which simplifies to

8m^2 + 6m = 0

. The non-zero solution is

m = -0.75

.

Step-by-Step Solution

1

Complete the square for the original circle equation

x^2 + y^2 - 12x - 4y + 31 = 0

.

(x - 6)^2 + (y - 2)^2 = 9

.

Grouping the

x

and

y

terms and completing the square yields the standard form of a circle, showing that the original center is

(6, 2)

and the radius

r = \sqrt{9} = 3

.

2

Apply the translation of

3

units right and

5

units down to the center

(6, 2)

.

New center is

(6 + 3, 2 - 5) = (9, -3)

.

Translating a point in the coordinate plane shifts the coordinates directly:

x

-coordinate increases by

3

and

y

-coordinate decreases by

5

. The radius remains

3

.

3

Set up the distance from the new center

(9, -3)

to the tangent line

y = mx

.

Distance equation:

\frac{|9m - (-3)|}{\sqrt{m^2 + 1}} = 3

.

The line

y = mx

can be rewritten as

mx - y = 0

. The perpendicular distance from the center to a tangent line must equal the radius of the circle, which is

3

.

4

Simplify the distance equation and solve for

m

.

|3m + 1| = \sqrt{m^2 + 1}

.

Factoring out

3

from the numerator and dividing both sides by

3

simplifies the equation before squaring.

5

Square both sides of the equation and solve the resulting quadratic equation.

2m(4m + 3) = 0 \implies m = 0

or

m = -0.75

.

Squaring both sides yields

(3m + 1)^2 = m^2 + 1 \implies 9m^2 + 6m + 1 = m^2 + 1 \implies 8m^2 + 6m = 0

. Since we are looking for a non-zero slope, the only valid solution is

m = -0.75

.

Key Concept

Using completing the square, coordinate translations, and the point-to-line distance formula to solve circle tangency problems

Question 2425Question →

A geometric sequence has a second term of $-\frac{3}{2}$ and a fifth term of $12$ . What is the eighth term of this sequence?

-96

-48

\frac{51}{2}

96

192

Show answer & explanation

Answer: $-96$

Answer

-96

The correct term is

-96

. First, find the common ratio

r

by taking the ratio of the fifth term to the second term:

\frac{a_5}{a_2} = \frac{a_1 r^4}{a_1 r} = r^3

. Substituting the given values,

r^3 = \frac{12}{-3/2} = -8

, which gives

r = -2

. To find the eighth term, multiply the fifth term by the common ratio cubed:

a_8 = a_5 r^3 = 12 \times (-2)^3 = 12 \times (-8) = -96

.

Step-by-Step Solution

1

Set up the ratio between the fifth term and the second term using the geometric sequence formula

a_n = a_1 r^{n-1}

.

\frac{a_5}{a_2} = \frac{a_1 r^4}{a_1 r} = r^3

This allows us to isolate the common ratio

r

without needing to calculate the first term

a_1

first.

2

Substitute the given values into the ratio and solve for

r

.

r^3 = \frac{12}{-3/2} = 12 \times \left(-\frac{2}{3}\right) = -8 \Rightarrow r = -2

Finding the common ratio is necessary to compute any subsequent terms in the sequence.

3

Use the common ratio to find the eighth term

a_8

by multiplying the fifth term

a_5

by

r^3

.

a_8 = a_5 r^{8-5} = 12 \times (-2)^3 = 12 \times (-8) = -96

Since

a_8 = a_5 r^3

, multiplying

12

by

-8

directly gives the eighth term.

Key Concept

Finding terms in a geometric sequence using the common ratio.

Question 2426Question →

Which inequality represents all real values of $x$ for which the inequality $\frac{5 - 2x}{3} \geq \frac{x - 6}{2}$ is true?

x \geq 4

x \leq \frac{11}{3}

x \leq 4

x \geq \frac{11}{3}

x \leq -\frac{8}{7}

Show answer & explanation

Answer: $x \leq 4$

Answer

The correct inequality is

x \leq 4

.

The correct inequality is

x \leq 4

. Multiplying both sides by the least common multiple,

6

, yields the inequality

2(5 - 2x) \geq 3(x - 6)

. Distributing the coefficients results in

10 - 4x \geq 3x - 18

. Gathering the variable terms by subtracting

3x

gives

10 - 7x \geq -18

. Subtracting

10

from both sides results in

-7x \geq -28

. Dividing both sides by

-7

and reversing the inequality sign results in the final solution

x \leq 4

.

Step-by-Step Solution

1

Multiply both sides of the inequality by

6

(the least common multiple of

2

and

3

) to eliminate the fractions.

2(5 - 2x) \geq 3(x - 6)

Eliminating denominators simplifies the linear inequality for solving.

2

Distribute the constants on both sides.

10 - 4x \geq 3x - 18

Expanding the terms allows combining like terms next.

3

Subtract

3x

from both sides of the inequality.

10 - 7x \geq -18

Grouping all variable terms on one side of the inequality.

4

Subtract

10

from both sides of the inequality.

-7x \geq -28

Isolating the variable term on the left side.

5

Divide both sides by

-7

and reverse the inequality sign.

x \leq 4

Dividing or multiplying both sides of an inequality by a negative number requires reversing the inequality sign direction.

Key Concept

Solving linear inequalities by clearing denominators and applying the sign-reversal rule when dividing by a negative number.

Estimated Time:1m 15s

Question 2427Question →

A triangle has vertices $K(-2, 3)$ , $L(-2, 1)$ , and $M(2, 1)$ in the standard $(x, y)$ coordinate plane. The triangle is first reflected across the line $y = x$ , and then the reflected image is translated $3$ units to the right and $4$ units down. What are the coordinates of the final image of vertex $K$ ?

(0, 2)

(1, -1)

(6, -6)

(0, -2)

(5, -1)

Show answer & explanation

Answer: $(6, -6)$

Answer

(6, -6)

The correct answer is

(6, -6)

. To find this, start with the coordinates of vertex

K(-2, 3)

. First, apply the reflection across the line

y = x

, which maps

(x, y) \rightarrow (y, x)

. This gives the intermediate coordinates

K'(3, -2)

. Second, apply the translation

3

units to the right and

4

units down, which maps

(x, y) \rightarrow (x + 3, y - 4)

. Adding

3

to the

x

-coordinate and subtracting

4

from the

y

-coordinate of the intermediate point yields the final coordinates

(3 + 3, -2 - 4) = (6, -6)

.

Step-by-Step Solution

1

Apply the reflection rule to the pre-image vertex

K(-2, 3)

K'(3, -2)

Reflecting a point

(x, y)

across the line

y = x

swaps its coordinates, mapping it to

(y, x)

.

2

Apply the translation rule to the reflected point

K'(3, -2)

K''(6, -6)

Translating a point

3

units to the right and

4

units down adds

3

to the

x

-coordinate and subtracts

4

from the

y

-coordinate.

Key Concept

Composite transformations in the coordinate plane

Question 2428Question →

A rectangular garden has a length of $3x^2$ meters and a width of $2x^3 - 5$ meters. A square storage shed built inside the garden has a side length of $x^2 - 3$ meters. Which of the following polynomials represents the area of the remaining garden space, in square meters, that is not covered by the shed?

6x^5 - x^4 - 15x^2 - 9

6x^6 - x^4 - 9x^2 - 9

6x^5 - x^4 - 21x^2 + 9

6x^5 - x^4 - 9x^2 - 9

6x^5 - x^4 - 9x^2 + 9

Show answer & explanation

Answer: $6x^5 - x^4 - 9x^2 - 9$

Answer

The polynomial

6x^5 - x^4 - 9x^2 - 9

represents the remaining area.

To find the remaining area, subtract the area of the square shed from the area of the rectangular garden. The area of the garden is

3x^2(2x^3 - 5) = 6x^5 - 15x^2

. The area of the shed is

(x^2 - 3)^2 = x^4 - 6x^2 + 9

. Subtracting the two gives

(6x^5 - 15x^2) - (x^4 - 6x^2 + 9) = 6x^5 - 15x^2 - x^4 + 6x^2 - 9 = 6x^5 - x^4 - 9x^2 - 9

, which represents the correct remaining garden space.

Step-by-Step Solution

1

Calculate the area of the rectangular garden.

\text{Area}_{\text{garden}} = 3x^2(2x^3 - 5) = 6x^5 - 15x^2

The area of a rectangle is the product of its length and width. Applying the distributive property and the exponent rule for multiplication (

x^a \cdot x^b = x^{a+b}

), we get

3x^2 \cdot 2x^3 = 6x^5

and

3x^2 \cdot (-5) = -15x^2

.

2

Calculate the area of the square storage shed.

\text{Area}_{\text{shed}} = (x^2 - 3)^2 = x^4 - 6x^2 + 9

The area of a square is the square of its side length. We expand

(x^2 - 3)^2

using the binomial square pattern

(a - b)^2 = a^2 - 2ab + b^2

, which gives

(x^2)^2 - 2(x^2)(3) + (-3)^2 = x^4 - 6x^2 + 9

.

3

Subtract the area of the shed from the area of the garden and simplify.

\text{Remaining Area} = 6x^5 - x^4 - 9x^2 - 9

We subtract the shed's area from the garden's area:

(6x^5 - 15x^2) - (x^4 - 6x^2 + 9)

. Distributing the negative sign gives

6x^5 - 15x^2 - x^4 + 6x^2 - 9

. Combining like terms and writing in descending order yields

6x^5 - x^4 - 9x^2 - 9

.

Key Concept

Operations on Polynomials

Question 2429Question →

In right triangle $ABC$ , the measure of $\angle B$ is $90^\circ$ , the measure of $\angle A$ is $60^\circ$ , and the hypotenuse $AC$ has a length of $20$ centimeters. Point $D$ lies on leg $BC$ such that the length of segment $BD$ is $2\sqrt{3}$ centimeters. A line segment $DE$ is drawn perpendicular to $AC$ such that $E$ lies on $AC$ . What is the length, in centimeters, of segment $AE$ ?

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By solving for the angles and side lengths of the two nested

30^\circ\text{-}60^\circ\text{-}90^\circ

right triangles, we find that the segment

BC

is

10\sqrt{3}

cm, making

DC = 8\sqrt{3}

cm. Using the ratio of sides for the smaller right triangle

DEC

, we find

EC = 12

cm, which leaves

AE = 20 - 12 = 8

cm.

Step-by-Step Solution

1

Determine the third angle of right triangle

ABC

.

\angle C = 30^\circ

The sum of angles in a triangle is

180^\circ

. Since

\angle B = 90^\circ

and

\angle A = 60^\circ

, we have

\angle C = 180^\circ - 90^\circ - 60^\circ = 30^\circ

.

2

Calculate the length of the side

BC

.

BC = 10\sqrt{3}

cm

In the

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle

ABC

, the side

BC

is opposite the

60^\circ

angle, so its length is the hypotenuse

AC

multiplied by

\sin(60^\circ)

or

\frac{\sqrt{3}}{2}

. Thus,

BC = 20 \times \frac{\sqrt{3}}{2} = 10\sqrt{3}

.

3

Find the length of segment

DC

.

DC = 8\sqrt{3}

cm

Since point

D

lies on segment

BC

, the length of

DC

is the total length of

BC

minus the length of

BD

. Since

BD = 2\sqrt{3}

, we have

DC = 10\sqrt{3} - 2\sqrt{3} = 8\sqrt{3}

.

4

Determine the properties of the right triangle

DEC

.

\triangle DEC

is a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle with hypotenuse

DC = 8\sqrt{3}

cm.

Since segment

DE

is perpendicular to

AC

,

\angle DEC = 90^\circ

. Triangle

DEC

shares the angle

\angle C = 30^\circ

with triangle

ABC

, which makes it a

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle where

DC

is the hypotenuse.

5

Calculate the length of segment

EC

.

EC = 12

cm

In the

30^\circ\text{-}60^\circ\text{-}90^\circ

triangle

DEC

, the leg

EC

is adjacent to the

30^\circ

angle, so its length is the hypotenuse

DC

multiplied by

\cos(30^\circ)

or

\frac{\sqrt{3}}{2}

. Thus,

EC = 8\sqrt{3} \times \frac{\sqrt{3}}{2} = 12

.

6

Calculate the length of segment

AE

.

AE = 8

cm

Since point

E

lies on segment

AC

, we can find

AE

by subtracting

EC

from

AC

. Thus,

AE = AC - EC = 20 - 12 = 8

.

Key Concept

Using the properties of

30^\circ\text{-}60^\circ\text{-}90^\circ

special right triangles to find missing lengths in composite geometric configurations.

Question 2430Question →

The lengths of the three sides of a triangle are in the ratio $3:4:x$ , where $x$ is an integer. If the perimeter of the triangle is $36$ centimeters, how many different possible values can $x$ have?

1

3

2

4

5

Show answer & explanation

Answer: 2

Answer

There are 2 possible integer values for

x

.

The correct answer is the value of 2. By expressing the side lengths as

3k

,

4k

, and

xk

where

k

must be a positive integer, the perimeter equation becomes

k(7 + x) = 36

. Solving for

7 + x

yields factors of

36

greater than

7

, which are

9, 12, 18,

and

36

. These correspond to

x

values of

2, 5, 11,

and

29

. Checking each set of side lengths against the Triangle Inequality Theorem shows that only the sets corresponding to

x = 2

(sides

12, 16, 8

) and

x = 5

(sides

9, 12, 15

) form valid triangles.

Step-by-Step Solution

1

Define the side lengths using a multiplier

k

.

Let the side lengths of the triangle be

3k

,

4k

, and

xk

for some positive multiplier

k

. Since all three side lengths must be integers, the difference between the first two sides,

4k - 3k = k

, must also be an integer. Thus,

k

must be a positive integer.

This establishes that the scaling factor

k

is a positive integer, allowing us to find discrete solutions.

2

Set up the perimeter equation and express

x

in terms of

k

.

The perimeter is the sum of the side lengths:

3k + 4k + xk = 36 \implies k(7 + x) = 36

. Since

k

and

x

are positive integers,

7 + x

must be a factor of

36

that is greater than

7

.

This constrains the possible values of

x

to the factors of

36

that are larger than

7

.

3

Find the potential values of

x

and their corresponding side lengths.

The factors of

36

greater than

7

are

9, 12, 18,

and

36

. This yields four potential cases:
- If

7+x = 9 \implies x = 2

, then

k = 4

, and the sides are

12, 16, 8

.
- If

7+x = 12 \implies x = 5

, then

k = 3

, and the sides are

9, 12, 15

.
- If

7+x = 18 \implies x = 11

, then

k = 2

, and the sides are

6, 8, 22

.
- If

7+x = 36 \implies x = 29

, then

k = 1

, and the sides are

3, 4, 29

.

This identifies all mathematically possible configurations before checking if they can physically form a triangle.

4

Apply the Triangle Inequality Theorem to each case.

The sum of the lengths of any two sides must be strictly greater than the length of the remaining side:
- For sides

12, 16, 8

:

12 + 8 = 20 > 16

(Valid).
- For sides

9, 12, 15

:

9 + 12 = 21 > 15

(Valid).
- For sides

6, 8, 22

:

6 + 8 = 14 < 22

(Invalid).
- For sides

3, 4, 29

:

3 + 4 = 7 < 29

(Invalid).
Only the cases where

x = 2

and

x = 5

form valid triangles.

This filters the candidate values of

x

to only those that can form a valid geometric triangle.

Key Concept

Triangle Inequality Theorem and Integer Ratio Constraints

Estimated Time:1m 30s

Question 2431Question →

A convex pentagon has four interior angles that measure $80^\circ$ , $110^\circ$ , $120^\circ$ , and $130^\circ$ . What is the degree measure of the fifth interior angle?

Show answer & explanation

Answer: 100

Answer

The degree measure of the fifth interior angle is

100^\circ

.

The sum of the interior angles of a pentagon (

n = 5

) is

(5 - 2) \times 180^\circ = 540^\circ

. The sum of the four given angles is

80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ

. The measure of the fifth angle is the difference between these two values:

540^\circ - 440^\circ = 100^\circ

.

Step-by-Step Solution

1

Find the sum of the interior angles of a convex pentagon.

The sum of the interior angles is

540^\circ

.

The sum of the interior angles of any

n

-sided convex polygon is calculated using the formula

(n - 2) \times 180^\circ

. For a pentagon,

n = 5

, which gives

(5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ

.

2

Sum the measures of the four given interior angles.

The sum of the four given angles is

440^\circ

.

Adding the given measures:

80^\circ + 110^\circ + 120^\circ + 130^\circ = 440^\circ

.

3

Subtract the sum of the four given angles from the total sum of the interior angles.

The measure of the fifth interior angle is

100^\circ

.

Subtracting the sum of the known angles from the total pentagon interior angle sum yields

540^\circ - 440^\circ = 100^\circ

.

Key Concept

The sum of the interior angles of an

n

-sided convex polygon is

(n-2) \times 180^\circ

.

Question 2432Question →

A right triangle has legs of length $5$ inches and $12$ inches. What is the length, in inches, of the hypotenuse of this triangle?

Show answer & explanation

Answer: 13

Answer

The length of the hypotenuse is 13 inches.

The Pythagorean theorem states that in any right triangle with legs

a

and

b

and hypotenuse

c

,

a^2 + b^2 = c^2

. Substituting the given values:

5^2 + 12^2 = 25 + 144 = 169

. Taking the square root of

169

gives the hypotenuse length of

13

.

Step-by-Step Solution

1

Identify the lengths of the two legs.

a = 5

,

b = 12

These are the given side lengths perpendicular to each other.

2

Apply the Pythagorean theorem

a^2 + b^2 = c^2

.

5^2 + 12^2 = 25 + 144 = 169

To find the square of the hypotenuse.

3

Solve for the hypotenuse

c

by taking the square root.

c = \sqrt{169} = 13

To find the side length of the hypotenuse.

Key Concept

Pythagorean Theorem

Question 2433Question →

A line is defined by the equation $y = 2x + k$ , where $k$ is a constant. A parabola is defined by the equation $y = x^2 - 4x + 14$ . If the line and the parabola intersect at exactly one point in the standard $(x, y)$ coordinate plane, what is the value of $k$ ?

Show answer & explanation

Answer: 5

Answer

The value of the constant

k

must be 5.

To find the intersection of the line and the parabola, set their equations equal to each other:

x^2 - 4x + 14 = 2x + k

. Rearranging this into standard quadratic form gives

x^2 - 6x + (14 - k) = 0

. For the system to have exactly one solution, the discriminant of this quadratic equation must be zero. The discriminant is

b^2 - 4ac = (-6)^2 - 4(1)(14 - k) = 36 - 56 + 4k = 4k - 20

. Setting

4k - 20 = 0

yields

k = 5

.

Step-by-Step Solution

1

Equate the linear and quadratic expressions to find their intersection.

x^2 - 4x + 14 = 2x + k

Setting the two equations equal to each other allows us to find the

x

-coordinates of any intersection points.

2

Rearrange the terms to write the equation in standard quadratic form,

ax^2 + bx + c = 0

.

x^2 - 6x + (14 - k) = 0

Subtracting

2x

and

k

from both sides simplifies the equation into a quadratic form where

a = 1

,

b = -6

, and

c = 14 - k

.

3

Apply the condition for exactly one intersection point by setting the discriminant to zero.

(-6)^2 - 4(1)(14 - k) = 0

A quadratic equation has exactly one real root if and only if its discriminant,

b^2 - 4ac

, is equal to zero.

4

Solve the linear equation for

k

.

k = 5

Expanding the equation yields

36 - 56 + 4k = 0

, which simplifies to

4k - 20 = 0

, leading to

k = 5

.

Key Concept

Determining the condition for a linear equation to be tangent to a quadratic equation by setting the discriminant of their intersection equation to zero.

Question 2434Question →

A custom printing company uses a linear relationship to determine the total cost of printing t-shirts. The table below displays the total cost, $C$ , in dollars, for printing $x$ t-shirts:

Number of Shirts ( $x$ )	Total Cost ( $C$ )
15	\$162.50
25	\$237.50
35	\$312.50

If the total cost consists of a one-time setup fee plus a constant charge per t-shirt, what is the one-time setup fee?

\$7.50

\$10.83

\$12.50

\$50.00

\$112.50

Show answer & explanation

Answer: \$50.00

Answer

The one-time setup fee is \$50.00.

The correct answer of \

50.00 is found by first determining the constant rate of change (slope) from the table, which is \$7.50 per shirt. Substituting this rate and the data point for 15 shirts into the linear model

C = mx + b

gives

162.50 = 7.50(15) + b

. Solving for the y-intercept

b

yields

50.00, which represents the one-time setup fee.

Step-by-Step Solution

1

Calculate the constant charge per t-shirt (the slope,

m

) by finding the change in cost divided by the change in the number of shirts between two data points.

m = \frac{237.50 - 162.50}{25 - 15} = \frac{75.00}{10} = 7.50

Since the relationship is linear, the rate of change (cost per shirt) remains constant.

2

Write the linear equation representing the total cost:

C = 7.50x + b

, where

b

represents the one-time setup fee.

C = 7.50x + b

This sets up the slope-intercept form of the linear relationship.

3

Substitute one of the points from the table, such as

(15, 162.50)

, into the equation to solve for

b

.

162.50 = 7.50(15) + b

162.50 = 112.50 + b

b = 50.00

Using a known coordinate point allows us to isolate and solve for the unknown y-intercept (the setup fee).

Key Concept

Linear Equations and Graphing

Question 2435Question →

In the standard $(x, y)$ coordinate plane, the center of a circle is located at the midpoint of the line segment with endpoints $A\left(\frac{1}{2}, 9\right)$ and $B\left(\frac{9}{2}, 3\right)$ . If the circle passes through the point $P\left(\frac{13}{2}, 9\right)$ , what is the radius of the circle?

6

7

5

25

2\sqrt{13}

Show answer & explanation

Answer: 5

Answer

The radius of the circle is 5.

The center of the circle is the midpoint of segment AB. Calculating the midpoint of

A(0.5, 9)

and

B(4.5, 3)

gives

M(2.5, 6)

. The radius is the distance from this center

M(2.5, 6)

to the given point

P(6.5, 9)

on the circle. The distance is

\sqrt{(6.5 - 2.5)^2 + (9 - 6)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

Step-by-Step Solution

1

Find the coordinates of the center of the circle by calculating the midpoint of segment AB.

Center

M = \left(\frac{1/2 + 9/2}{2}, \frac{9 + 3}{2}\right) = \left(\frac{5}{2}, 6\right) = (2.5, 6)

.

The center of the circle is located at the midpoint of the line segment AB.

2

Use the distance formula to find the distance between the center

M(2.5, 6)

and the point

P(6.5, 9)

on the circle.

r = \sqrt{(6.5 - 2.5)^2 + (9 - 6)^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5

.

The radius of a circle is the distance from its center to any point on its boundary.

Key Concept

Using the midpoint formula to locate the center of a circle, and then applying the distance formula between the center and a boundary point to find the radius.

Alternative Method

You can visualize the horizontal and vertical distances on a coordinate plane. The horizontal distance between the x-coordinates of the center

(2.5, 6)

and the point

(6.5, 9)

is

6.5 - 2.5 = 4

. The vertical distance is

9 - 6 = 3

. Recognizing the 3-4-5 right triangle triplet immediately gives the distance (hypotenuse) as 5.

Estimated Time:1m 15s

Question 2436Question →

A $13$ -foot ladder is leaning against a flat vertical wall. The base of the ladder is placed $5$ feet away from the bottom of the wall. How many feet up the wall does the ladder reach?

\sqrt{8}

8

12

\sqrt{194}

18

Show answer & explanation

Answer: $12$

Answer

The height the ladder reaches is

12

feet.

The correct answer is

12

feet. The ladder, wall, and ground form a right triangle where the ladder is the hypotenuse (

13

feet) and the distance along the ground is one leg (

5

feet). By the Pythagorean theorem, the height up the wall,

b

, satisfies

5^2 + b^2 = 13^2

. Solving for

b

gives

b^2 = 169 - 25 = 144

, so

b = 12

.

Step-by-Step Solution

1

Identify the hypotenuse and the given leg from the word problem description.

The ladder length is the hypotenuse (

c = 13

), and the distance from the wall is one of the legs (

a = 5

).

The ladder forms the diagonal side opposite the right angle formed by the vertical wall and the ground.

2

Set up the Pythagorean theorem to find the unknown leg length.

5^2 + b^2 = 13^2

.

The Pythagorean theorem (

a^2 + b^2 = c^2

) relates the side lengths of any right triangle.

3

Solve for the unknown leg length

b

by simplifying terms and taking the square root.

25 + b^2 = 169 \implies b^2 = 144 \implies b = \sqrt{144} = 12

.

Subtracting the square of the known leg from the square of the hypotenuse isolates the squared unknown leg, which can then be solved by finding its square root.

Key Concept

Using the Pythagorean theorem to find an unknown leg of a right triangle when the hypotenuse and one leg are known.

Alternative Method

Recognizing that

5

and

13

are part of the common Pythagorean triple

5

-

12

-

13

allows you to immediately identify the missing leg as

12

without performing calculations.

Estimated Time:45s

Question 2437Question →

An arithmetic sequence has a first term of $2$ and a common difference of $d$ . A geometric sequence has a first term of $4$ and a common ratio of $r$ . The third term of the arithmetic sequence is equal to the third term of the geometric sequence. If $d = r$ and $d \neq 1$ , what is the value of $d$ ?

2

\frac{1}{2}

-1

-\frac{1}{2}

\frac{1}{3}

Show answer & explanation

Answer: $-\frac{1}{2}$

Answer

-\frac{1}{2}

The correct answer is

-\frac{1}{2}

. The third term of the arithmetic sequence is

a_3 = a_1 + 2d = 2 + 2d

. The third term of the geometric sequence is

g_3 = g_1 r^2 = 4r^2

. Given that

d = r

, we set the two terms equal:

2 + 2d = 4d^2

. Rearranging and dividing by

2

yields

2d^2 - d - 1 = 0

, which factors as

(2d + 1)(d - 1) = 0

. Since the problem specifies that

d \neq 1

, the only valid solution is

d = -\frac{1}{2}

.

Step-by-Step Solution

1

Write the formulas for the third term of both the arithmetic and geometric sequences.

For the arithmetic sequence:

a_3 = a_1 + (3 - 1)d = 2 + 2d

. For the geometric sequence:

g_3 = g_1 \cdot r^{3 - 1} = 4r^2

.

This establishes the algebraic expressions for the third terms using the given first terms.

2

Substitute

d

for

r

and set the two expressions equal to each other.

Since

d = r

, we substitute

d

into the geometric term to get

g_3 = 4d^2

. Setting them equal gives

2 + 2d = 4d^2

.

The problem states that the third terms are equal and that the common difference equals the common ratio.

3

Solve the quadratic equation

4d^2 - 2d - 2 = 0

for

d

.

Divide the equation by

2

to get

2d^2 - d - 1 = 0

. Factoring this gives

(2d + 1)(d - 1) = 0

. The roots are

d = -\frac{1}{2}

and

d = 1

. Since the problem specifies

d \neq 1

, we have

d = -\frac{1}{2}

.

Solving the quadratic equation gives the possible values of the common difference, and the constraint rules out

d = 1

.

Key Concept

Relating arithmetic and geometric sequence terms and solving the resulting quadratic equation.

Alternative Method

Instead of factoring, the quadratic formula can be used to solve

2d^2 - d - 1 = 0

:

d = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(2)(-1)}}{2(2)} = \frac{1 \pm \sqrt{9}}{4} = \frac{1 \pm 3}{4}

. This yields

d = 1

and

d = -\frac{1}{2}

. Since the problem specifies

d \neq 1

, the only valid solution is

d = -\frac{1}{2}

.

Estimated Time:1m 30s

Question 2438Question →

A line segment with endpoints $C(2, 3)$ and $D(6, 3)$ is plotted in the standard $(x, y)$ coordinate plane. If this segment is rotated $90^\circ$ clockwise about the origin and then translated $4$ units up, what are the coordinates of the midpoint of the resulting segment?

(3, -8)

(7, -4)

(3, 0)

(-3, 8)

(4, 7)

Show answer & explanation

Answer: $(3, 0)$

Answer

(3, 0)

The correct answer is obtained by first calculating the midpoint of the original segment

CD

, which yields

(4, 3)

. Applying a

90^\circ

clockwise rotation about the origin maps the point

(x, y)

to

(y, -x)

, transforming

(4, 3)

to

(3, -4)

. Translating this intermediate point 4 units up adds 4 to its y-coordinate, resulting in

(3, 0)

.

Step-by-Step Solution

1

Find the midpoint of the original line segment

CD

using the midpoint formula

M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)

.

M = \left(\frac{2 + 6}{2}, \frac{3 + 3}{2}\right) = (4, 3)

Since translations and rotations are rigid transformations, the midpoint of the transformed segment is the image of the midpoint of the original segment.

2

Apply the rotation of

90^\circ

clockwise about the origin to the midpoint

M(4, 3)

. The rule for a clockwise rotation of

90^\circ

is

(x, y) \rightarrow (y, -x)

.

M'(3, -4)

Rotating a point

(x, y)

by

90^\circ

clockwise maps it to

(y, -x)

.

3

Translate the point

M'(3, -4)

by

4

units up. The rule for translating a point

d

units up is

(x, y) \rightarrow (x, y + d)

.

M''(3, -4 + 4) = (3, 0)

A vertical translation upward increases the y-coordinate of the point by the given number of units.

Key Concept

Transformations in the Coordinate Plane

Question 2439Question →

What is the smallest integer value of $y$ that satisfies the inequality $5(2 - y) < 3(y - 6)$ ?

Show answer & explanation

Answer: 4

Answer

The smallest integer value of

y

that satisfies the inequality is

4

.

Evaluating the inequality leads to

y > 3.5

. The smallest integer greater than

3.5

is

4

. Substituting

y = 4

into the original inequality gives

5(2 - 4) < 3(4 - 6) \implies -10 < -6

, which is true. Substituting the next smallest integer,

3

, gives

-5 < -9

, which is false.

Step-by-Step Solution

1

Distribute the coefficients to the terms inside the parentheses on both sides.

10 - 5y < 3y - 18

To clear the parentheses and simplify the terms.

2

Subtract

3y

and

10

from both sides of the inequality to group the variable terms on one side and constants on the other.

-8y < -28

To isolate the variable term.

3

Divide both sides by

-8

and reverse the inequality sign because of division by a negative number.

y > 3.5

To solve the inequality for

y

.

4

Identify the smallest integer that satisfies the inequality

y > 3.5

.

4

To find the smallest integer value greater than

3.5

.

Key Concept

Solving multi-step linear inequalities involving distribution and division by a negative number.

Alternative Method

Instead of subtracting variables to the left, we can add

5y

to both sides to keep the variable coefficient positive:

10 - 5y < 3y - 18 \implies 10 < 8y - 18 \implies 28 < 8y \implies y > 3.5

. This avoids the need to divide by a negative number and flip the sign, reducing the risk of a sign-flip error.

Estimated Time:1m 0s

Question 2440Question →

If $k$ is a non-zero constant, for what value of $k$ does the quadratic equation $(x - 3)^2 = kx$ have exactly one real solution?

6

-3

-12

-6

12

Show answer & explanation

Answer: $-12$

Answer

-12

Expanding

(x - 3)^2

yields

x^2 - 6x + 9 = kx

. Subtracting

kx

from both sides and grouping like terms gives

x^2 - (6 + k)x + 9 = 0

. For a quadratic equation to have exactly one real solution, its discriminant must be equal to zero. Thus, we set

(-(6+k))^2 - 4(1)(9) = 0

, which simplifies to

(6+k)^2 - 36 = 0

. Taking the square root of both sides gives

6 + k = 6

or

6 + k = -6

. Solving these equations gives

k = 0

or

k = -12

. Since the problem specifies that

k

is a non-zero constant, the correct value is

-12

.

Step-by-Step Solution

1

Expand the squared binomial on the left side of the equation.

x^2 - 6x + 9 = kx

Expanding the binomial

(x - 3)^2

allows us to rewrite the equation in a form where we can group terms.

2

Move

kx

to the left side and group the

x

terms to write the equation in standard quadratic form

ax^2 + bx + c = 0

.

x^2 - (6 + k)x + 9 = 0

, where

a = 1

,

b = -(6 + k)

, and

c = 9

.

The coefficients

a

,

b

, and

c

must be identified from the standard quadratic form to calculate the discriminant.

3

Set the discriminant

b^2 - 4ac

equal to

0

and solve for

k

.

(-(6+k))^2 - 4(1)(9) = 0 \implies (6+k)^2 - 36 = 0 \implies (6+k)^2 = 36 \implies 6+k = \pm 6

. This yields

k = 0

or

k = -12

. Since

k

is non-zero,

k = -12

.

A quadratic equation has exactly one real solution if and only if its discriminant is equal to zero.

Key Concept

Using the discriminant (

b^2 - 4ac = 0

) to determine when a quadratic equation has exactly one real solution.

Alternative Method

Alternatively, one can recognize that the equation

(x-3)^2 = kx

can be written as

x^2 - (6+k)x + 9 = 0

. For a quadratic equation with a leading coefficient of

1

and a constant term of

9

to have exactly one real solution, it must be a perfect square trinomial. A perfect square trinomial of the form

x^2 + bx + 9

must have

b = \pm 6

. Setting the middle coefficient equal to these values gives

-(6+k) = 6 \implies k = -12

or

-(6+k) = -6 \implies k = 0

. Since

k

is non-zero,

k = -12

.

Estimated Time:1m 30s

Answer

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