All practice questions

If $2^{2x+2} - 2^{2x} = 12$ for some real number $x$ , what is the value of $2^{4x}$ ?

Show answer & explanation

Answer: 16

Answer

16

By applying the product rule of exponents, the expression

2^{2x+2}

can be rewritten as

2^{2x} \cdot 2^2

, or

4 \cdot 2^{2x}

. Substituting this into the given equation yields

4 \cdot 2^{2x} - 2^{2x} = 12

. Factoring out

2^{2x}

gives

2^{2x}(4 - 1) = 12

, which simplifies to

3 \cdot 2^{2x} = 12

. Dividing both sides of the equation by 3 results in

2^{2x} = 4

. Since

2^{4x}

can be written as

(2^{2x})^2

, substituting 4 for

2^{2x}

gives

4^2 = 16

. Alternatively, solving

2^{2x} = 4

gives

2x = 2

, which means

x = 1

. Substituting

x = 1

into

2^{4x}

yields

2^{4(1)} = 2^4 = 16

.

Step-by-Step Solution

1

Rewrite

2^{2x+2}

using exponent properties.

2^{2x+2} = 2^{2x} \cdot 2^2 = 4 \cdot 2^{2x}

To express both exponential terms with the same base power, allowing them to be factored or combined.

2

Substitute this back into the equation and factor out the common term

2^{2x}

.

2^{2x}(4 - 1) = 12 \implies 3 \cdot 2^{2x} = 12

To isolate the exponential expression.

3

Solve for

2^{2x}

by dividing both sides of the equation by 3.

2^{2x} = 4

To find the value of the exponential term.

4

Express

2^{4x}

in terms of

2^{2x}

and evaluate.

2^{4x} = (2^{2x})^2 = 4^2 = 16

To find the final requested value using the exponent rule

(a^m)^n = a^{mn}

.

Key Concept

Factoring exponential equations and applying power of a power exponent rules

Question 1542Question →

A projectile is launched from a platform. Its height $h(t)$ , in meters, $t$ seconds after launch is modeled by the function $h(t) = -5t^2 + bt + 12$ , where $b$ is a positive constant. If the projectile reaches a maximum height of $32$ meters, what is the value of $b$ ?

Show answer & explanation

Answer: 20

Answer

The correct answer is 20.

To find the maximum height of the projectile, we locate the vertex of the quadratic function

h(t) = -5t^2 + bt + 12

. The

t

-coordinate of the vertex is given by

t = -\frac{B}{2A} = -\frac{b}{2(-5)} = \frac{b}{10}

. Substituting

t = \frac{b}{10}

into

h(t)

and setting the height to

32

meters gives

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

. Simplifying this equation results in

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which reduces to

\frac{b^2}{20} = 20

. Solving for

b

gives

b^2 = 400

. Because

b

must be a positive constant,

b = 20

.

Step-by-Step Solution

1

Identify the time

t

at which the maximum height occurs using the vertex formula

t = -\frac{B}{2A}

for the quadratic function

h(t) = At^2 + Bt + C

.

t = \frac{b}{10}

The vertex of a downward-opening parabola represents its maximum value. For

h(t) = -5t^2 + bt + 12

, the coefficients are

A = -5

and

B = b

.

2

Substitute the time

t = \frac{b}{10}

back into the height function

h(t)

and set the expression equal to the maximum height of

32

meters.

-5\left(\frac{b}{10}\right)^2 + b\left(\frac{b}{10}\right) + 12 = 32

At the maximum height, the height of the projectile is

32

meters, which corresponds to the value of the function at the vertex time.

3

Simplify the equation to solve for

b^2

.

\frac{b^2}{20} = 20

Squaring the fraction yields

-5\left(\frac{b^2}{100}\right) + \frac{b^2}{10} + 12 = 32

. Simplifying the coefficients leads to

-\frac{b^2}{20} + \frac{b^2}{10} = 20

, which simplifies to

\frac{b^2}{20} = 20

.

4

Solve the equation for

b

.

b = 20

Multiplying both sides by

20

gives

b^2 = 400

. Taking the square root of both sides gives

b = \pm 20

. Since

b

is specified to be a positive constant,

b = 20

.

Key Concept

Finding the vertex of a quadratic function to determine maximum values in a real-world context.

Estimated Time:2m 0s

Question 1543Question →

For a constant $k$ , the quadratic equation $x^2 - 2kx + k^2 - 4k + 3 = 0$ has two distinct real solutions. If both solutions are positive, which of the following describes all possible values of $k$ ?

k > 3

or

\frac{3}{4} < k < 1

k > 3

k > \frac{3}{4}

k < 1

or

k > 3

Show answer & explanation

Answer: $k > 3$ or $\frac{3}{4} < k < 1$

Answer

k > 3

or

\frac{3}{4} < k < 1

The correct answer is the option that includes both the interval between

\frac{3}{4}

and

1

and the interval greater than

3

. To ensure the quadratic equation has two distinct real solutions, the discriminant must be positive, which yields

k > \frac{3}{4}

. To ensure both solutions are positive, their sum and product must both be positive. The sum

2k > 0

requires

k > 0

. The product

k^2 - 4k + 3 > 0

requires

k < 1

or

k > 3

. The intersection of all these inequalities is

\frac{3}{4} < k < 1

or

k > 3

.

Step-by-Step Solution

1

Determine the condition for two distinct real solutions using the discriminant.

k > \frac{3}{4}

For the quadratic equation

x^2 - 2kx + k^2 - 4k + 3 = 0

to have two distinct real solutions, its discriminant

\Delta

must be strictly greater than

0

. Here,

a = 1

,

b = -2k

, and

c = k^2 - 4k + 3

. Thus,

\Delta = (-2k)^2 - 4(1)(k^2 - 4k + 3) = 4k^2 - 4k^2 + 16k - 12 = 16k - 12

. Setting

\Delta > 0

gives

16k - 12 > 0

, which simplifies to

k > \frac{3}{4}

.

2

Set up conditions for both solutions to be positive using the sum and product of the roots.

2k > 0

and

k^2 - 4k + 3 > 0

If a quadratic equation has two positive solutions, their sum must be positive and their product must also be positive. By Vieta's formulas, the sum of the solutions is

-\frac{b}{a} = 2k

and the product of the solutions is

\frac{c}{a} = k^2 - 4k + 3

.

3

Solve the inequalities for the sum and product of the roots.

k > 0

and either

k < 1

or

k > 3

The sum condition

2k > 0

simplifies to

k > 0

. The product condition

k^2 - 4k + 3 > 0

factors as

(k-1)(k-3) > 0

, which is satisfied when

k < 1

or

k > 3

.

4

Find the intersection of all the derived inequalities.

\frac{3}{4} < k < 1

or

k > 3

We must find the values of

k

that satisfy all three conditions:

k > \frac{3}{4}

,

k > 0

, and (

k < 1

or

k > 3

). Combining

k > \frac{3}{4}

and

k > 0

yields

k > \frac{3}{4}

. Combining

k > \frac{3}{4}

with the constraint that

k

must be less than

1

or greater than

3

gives the combined solution of

\frac{3}{4} < k < 1

or

k > 3

.

Key Concept

Using the discriminant and Vieta's formulas to determine the signs of the roots of a quadratic equation dependent on a parameter.

Alternative Method

Alternatively, you can write the solutions using the quadratic formula:

x = \frac{2k \pm \sqrt{16k - 12}}{2} = k \pm \sqrt{4k - 3}

. For both roots to be positive and distinct, we first need the term under the radical to be positive, so

4k - 3 > 0 \implies k > \frac{3}{4}

. Then, we need the smaller root to be positive:

k - \sqrt{4k - 3} > 0 \implies k > \sqrt{4k - 3}

. Since

k > \frac{3}{4} > 0

, we can square both sides:

k^2 > 4k - 3 \implies k^2 - 4k + 3 > 0 \implies (k - 1)(k - 3) > 0

. This yields

k < 1

or

k > 3

. Combining this with

k > \frac{3}{4}

gives the same result:

\frac{3}{4} < k < 1

or

k > 3

.

Estimated Time:3m 0s

Question 1544Question →

If $3^{x+4} = 9^x$ , what is the value of $x$ ?

4

2

-4

\frac{4}{3}

Show answer & explanation

Answer: 4

Answer

4

To solve the equation

3^{x+4} = 9^x

, we rewrite the base 9 as

3^2

, which gives

3^{x+4} = (3^2)^x

. Applying the exponent rule

(a^m)^n = a^{mn}

, we get

3^{x+4} = 3^{2x}

. Since the bases are now the same, we can set the exponents equal to each other:

x+4 = 2x

. Subtracting

x

from both sides yields the solution

4

.

Step-by-Step Solution

1

Express both sides of the equation with a common base.

Since

9 = 3^2

, the equation

3^{x+4} = 9^x

can be rewritten as

3^{x+4} = (3^2)^x

. Using the power of a power rule, this becomes

3^{x+4} = 3^{2x}

.

To solve an exponential equation, it is helpful to have the same base on both sides so that the exponents can be equated.

2

Set the exponents equal to each other.

x + 4 = 2x

Since the bases are equal and positive (and not equal to 1), their exponents must be equal.

3

Solve the linear equation for

x

.

4 = x

(or

x = 4

)

Subtract

x

from both sides of the equation to isolate the variable.

Key Concept

Solving exponential equations by expressing terms with a common base and equating their exponents.

Estimated Time:45s

Question 1545Question →

A point $(x, y)$ in the coordinate plane satisfies the system of inequalities below.

y \geq 2x - 4

y \leq -x + 5

x \geq 0

y \geq 0

What is the maximum possible value of $x$ for this point?

Show answer & explanation

Answer: 3

Answer

The maximum possible value of

x

that satisfies the system of inequalities is 3.

To find the maximum possible value of

x

for a point satisfying the system of inequalities, we identify the vertices of the region. The upper-right boundary of the region is formed by the intersection of the lines

y = 2x - 4

and

y = -x + 5

. Solving

2x - 4 = -x + 5

gives

3x = 9 \implies x = 3

. At this point,

y = -3 + 5 = 2

, which satisfies the constraints

x \geq 0

and

y \geq 0

. The other boundary vertices of the region are

(0, 0)

,

(2, 0)

, and

(0, 5)

. Comparing the x-coordinates of these vertices (

0

,

2

, and

3

), we see that the maximum possible value of

x

is

3

.

Step-by-Step Solution

1

Find the intersection point of the boundary lines

y = 2x - 4

and

y = -x + 5

.

x = 3

Setting the two boundary line equations equal to each other (

2x - 4 = -x + 5

) allows us to find the x-coordinate where the boundaries cross.

2

Substitute

x = 3

back into one of the boundary equations to find the y-coordinate.

y = 2

This yields the intersection point

(3, 2)

which lies on both boundary lines.

3

Verify that the intersection point

(3, 2)

satisfies the other constraints:

x \geq 0

and

y \geq 0

.

3 \geq 0

and

2 \geq 0

(both true)

The point must lie within the first quadrant to be a valid solution.

4

Determine the remaining boundary vertices of the solution region in the first quadrant.

Vertices are

(0, 0)

,

(0, 5)

,

(2, 0)

, and

(3, 2)

.

Comparing all vertices will confirm if

(3, 2)

indeed provides the maximum value of

x

.

Key Concept

Systems of Linear Inequalities in Two Variables

Question 1546Question →

A toy rocket is launched vertically upward from ground level. Its height, $h$ , in meters, $t$ seconds after launch is modeled by the equation $h(t) = -5t^2 + 40t$ . How many seconds after launch does the rocket reach its maximum height?

8

4

80

20

Show answer & explanation

Answer: 4

Answer

4 seconds

The height of the rocket is modeled by the quadratic function

h(t) = -5t^2 + 40t

. Because the coefficient of

t^2

is negative (

a = -5

), the graph of this function is a parabola that opens downward, meaning its vertex represents the maximum height. The time

t

at which the vertex occurs is given by

t = -\frac{b}{2a}

. Substituting

a = -5

and

b = 40

gives

t = -\frac{40}{2(-5)} = 4

seconds.

Step-by-Step Solution

1

Identify the coefficients of the quadratic equation.

a = -5

,

b = 40

, and

c = 0

The equation is given in standard quadratic form

h(t) = at^2 + bt + c

.

2

Apply the vertex formula to find the time at which the maximum height occurs.

t = -\frac{b}{2a}

For a downward-opening parabola, the maximum value occurs at the vertex.

3

Substitute the values of

a

and

b

and simplify.

t = -\frac{40}{2(-5)} = 4

This calculation yields the time in seconds when the rocket reaches its maximum height.

Key Concept

Finding the vertex of a quadratic function to determine the maximum or minimum value in context.

Alternative Method

Find the times when the rocket is at ground level by solving

h(t) = 0

. Factoring

-5t^2 + 40t = 0

gives

-5t(t - 8) = 0

, so the rocket is on the ground at

t = 0

seconds and

t = 8

seconds. Since a parabola is symmetric, the maximum height must occur exactly halfway between the launch and landing times, which is

\frac{0 + 8}{2} = 4

seconds.

Estimated Time:45s

Question 1547Question →

A circle in the $xy$ -plane is defined by the equation $(x - 3)^2 + (y - 2)^2 = 13$ . The line $y = 3x$ intersects the circle at the origin $(0, 0)$ and at a second point $P$ . What is the $y$ -coordinate of point $P$ ?

\frac{9}{5}

\frac{11}{5}

\frac{27}{5}

-\frac{9}{5}

Show answer & explanation

Answer: $\frac{27}{5}$

Answer

The y-coordinate of point P is

\frac{27}{5}

.

Substituting the linear relationship

y = 3x

into the circle equation yields a quadratic equation in terms of

x

. Solving this equation gives the

x

-coordinates of the intersection points as

x = 0

and

x = \frac{9}{5}

. Since point

P

is not the origin, its

x

-coordinate is

\frac{9}{5}

. Substituting this back into

y = 3x

gives the

y

-coordinate of point

P

as

\frac{27}{5}

.

Step-by-Step Solution

1

Substitute the linear equation

y = 3x

into the equation of the circle.

(x - 3)^2 + (3x - 2)^2 = 13

Substitution reduces the system of two equations with two variables to a single quadratic equation with one variable.

2

Expand and simplify the equation.

10x^2 - 18x = 0

Expanding

(x - 3)^2

to

x^2 - 6x + 9

and

(3x - 2)^2

to

9x^2 - 12x + 4

, then combining like terms and subtracting 13 from both sides, isolates the quadratic terms.

3

Solve for

x

by factoring.

2x(5x - 9) = 0

, which gives

x = 0

or

x = \frac{9}{5}

.

The roots of the factored equation represent the x-coordinates of the two intersection points.

4

Find the y-coordinate of the second point

P

using

y = 3x

.

y = 3 \left(\frac{9}{5}\right) = \frac{27}{5}

Since point

P

is not the origin, its x-coordinate must be

\frac{9}{5}

. Substituting this into the linear equation gives its corresponding y-coordinate.

Key Concept

Solving nonlinear systems of equations involving a circle and a line by substitution.

Question 1548Question →

The quadratic equation $3x^2 - 12x + c = 0$ , where $c$ is a constant, has two real solutions, $r$ and $s$ . If $\frac{1}{r} + \frac{1}{s} = \frac{3}{2}$ , what is the value of $c$ ?

-8

8

18

24

Show answer & explanation

Answer: 8

Answer

8

The correct answer is 8. By Vieta's formulas, the sum of the solutions is

r + s = -\frac{-12}{3} = 4

and their product is

rs = \frac{c}{3}

. The given equation

\frac{1}{r} + \frac{1}{s} = \frac{3}{2}

can be rewritten as

\frac{r+s}{rs} = \frac{3}{2}

. Substituting the expressions for the sum and product gives

\frac{4}{c/3} = \frac{12}{c} = \frac{3}{2}

. Solving this equation yields

c = 8

. Checking the discriminant of the resulting quadratic equation

3x^2 - 12x + 8 = 0

gives

\Delta = (-12)^2 - 4(3)(8) = 48

, which is positive and confirms the existence of two real solutions.

Step-by-Step Solution

1

Use Vieta's formulas to express the sum and product of the solutions in terms of the coefficients of the quadratic equation.

The sum of the solutions is

r + s = -\frac{b}{a} = -\frac{-12}{3} = 4

, and the product of the solutions is

rs = \frac{c}{a} = \frac{c}{3}

.

Vieta's formulas relate the sum and product of the roots of a quadratic equation

ax^2 + bx + c = 0

directly to its coefficients.

2

Express the given algebraic relation

\frac{1}{r} + \frac{1}{s}

in terms of

r+s

and

rs

by finding a common denominator.

\frac{1}{r} + \frac{1}{s} = \frac{s + r}{rs} = \frac{r + s}{rs}

.

Finding a common denominator allows us to rewrite the sum of reciprocals using the sum and product of the solutions.

3

Substitute the sum and product from Step 1 into the rewritten equation and solve for

c

.

Substituting the values gives

\frac{4}{\frac{c}{3}} = \frac{3}{2}

, which simplifies to

\frac{12}{c} = \frac{3}{2}

. Solving for

c

yields

3c = 24

, or

c = 8

.

This substitution turns the given relation into a single linear equation in terms of the variable

c

.

4

Verify that

c = 8

yields two real solutions by checking the discriminant of the quadratic equation.

The discriminant is

\Delta = b^2 - 4ac = (-12)^2 - 4(3)(8) = 144 - 96 = 48

. Since

\Delta > 0

, there are indeed two real solutions.

The problem states that the equation has two real solutions, so we must confirm that

c = 8

satisfies this condition.

Key Concept

Applying Vieta's formulas and algebraic manipulation to solve for constants in a quadratic equation.

Estimated Time:2m 0s

Question 1549Question →

\begin{aligned} y &= x^2 - x - 7 \\ 2x - y &= -3 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $x < 0$ , what is the value of $x + y$ ?

18

-3

-2

0

Show answer & explanation

Answer: -3

Answer

The correct answer is -3.

The correct answer is

-3

. To find this, substitute the expression for

y

from the first equation into the second equation, which yields

2x - (x^2 - x - 7) = -3

. Distributing the negative sign and combining like terms gives

-x^2 + 3x + 7 = -3

. Setting the equation to zero results in

x^2 - 3x - 10 = 0

. Factoring the quadratic expression gives

(x - 5)(x + 2) = 0

, which means

x = 5

or

x = -2

. Since the question specifies that

x < 0

, we select

x = -2

. Substituting

x = -2

back into the linear equation gives

y = 2(-2) + 3 = -1

. Finally, calculating

x + y

gives

-2 + (-1) = -3

.

Step-by-Step Solution

1

Substitute the expression for

y

from the first equation into the second equation.

2x - (x^2 - x - 7) = -3

This allows us to eliminate

y

and solve for

x

in a single variable quadratic equation.

2

Distribute the negative sign and simplify the equation.

-x^2 + 3x + 7 = -3

Expanding the parentheses correctly preserves the signs of the quadratic terms.

3

Rearrange the terms to set the quadratic equation to zero.

x^2 - 3x - 10 = 0

Writing the equation in standard form (

ax^2 + bx + c = 0

) allows us to factor it.

4

Factor the quadratic expression.

(x - 5)(x + 2) = 0

Finding factors that multiply to

-10

and add to

-3

gives us the solutions for

x

.

5

Solve for

x

and apply the constraint

x < 0

.

x = -2

The equation yields

x = 5

or

x = -2

. The constraint

x < 0

means we must choose

x = -2

.

6

Substitute

x = -2

back into one of the original equations to solve for

y

.

y = 2(-2) + 3 = -1

We need the corresponding

y

-value to find the value of

x + y

.

7

Calculate the value of

x + y

.

x + y = -2 + (-1) = -3

This provides the final required value specified in the question.

Key Concept

Solving a system of one linear equation and one quadratic equation using substitution.

Question 1550Question →

\begin{aligned} x - y &= 3 \\ y &= x^2 - 7x + 12 \end{aligned}

If $(x, y)$ is a solution to the system of equations above and $y > 0$ , what is the value of $xy$ ?

Show answer & explanation

Answer: 10

Answer

10

By substituting the linear equation

y = x - 3

into the quadratic equation, we obtain a single quadratic equation

x^2 - 8x + 15 = 0

. Factoring this equation yields the solutions

x = 3

and

x = 5

. The corresponding

y

-coordinates are

y = 0

and

y = 2

, respectively. Since the problem specifies that

y > 0

, we choose the solution

(5, 2)

. The product of the coordinates is

5 \cdot 2 = 10

.

Step-by-Step Solution

1

Express

y

in terms of

x

from the linear equation.

y = x - 3

This allows for substitution into the second equation.

2

Substitute

y = x - 3

into the quadratic equation.

x - 3 = x^2 - 7x + 12

To create a single quadratic equation in terms of

x

.

3

Rearrange the equation into standard quadratic form

ax^2 + bx + c = 0

.

x^2 - 8x + 15 = 0

Subtracting

x

and adding

3

to both sides enables factoring.

4

Factor the quadratic expression.

(x - 3)(x - 5) = 0

To find the roots of the equation, which are

x = 3

and

x = 5

.

5

Find the corresponding

y

-values for each

x

-value.

For

x = 3

,

y = 3 - 3 = 0

. For

x = 5

,

y = 5 - 3 = 2

.

To obtain the complete coordinate pairs of the intersection points.

6

Apply the constraint

y > 0

to select the correct solution pair.

The solution

(5, 2)

is selected since its

y

-coordinate is greater than

0

.

The other solution,

(3, 0)

, has

y = 0

, which does not satisfy the constraint

y > 0

.

7

Calculate the value of

xy

for the chosen solution.

xy = 5 \cdot 2 = 10

To find the final requested value.

Key Concept

Solving a nonlinear system of equations containing a linear equation and a quadratic equation by substitution.

Estimated Time:1m 30s

Question 1551Question →

The following passage is adapted from an academic text about deep-sea biology. Which punctuation mark should be placed in the blank to grammatically connect the items in the list?

Fill in the blanks below

In her analysis of hydrothermal vent ecosystems, marine biologist Dr. Elena Vance categorized the resident fauna into three zones based on temperature tolerance: the vent opening, which supports chemosynthetic bacteria in extreme heat the surrounding field, which is inhabited by tube worms and crabs; and the peripheral zone, where scavengers feed on falling organic debris.

Show answer & explanation

Answer

A semicolon (;) is the correct punctuation mark because it is used to separate items in a complex list where the items already contain internal punctuation (commas).

A semicolon (;) is required to separate the items in this list because the individual items contain internal commas (e.g., 'the vent opening, which supports...'). In Standard English, semicolons function as 'super-commas' in complex lists to prevent reader confusion and clearly indicate the division between each item.

Step-by-Step Solution

1

Analyze the structure of the sentence and the list.

The sentence contains a list of three distinct zones: the vent opening, the surrounding field, and the peripheral zone.

Identifying the list structure helps determine the appropriate punctuation for separating the items.

2

Examine the items in the list for internal punctuation.

Each of the listed items contains explanatory clauses set off by commas (e.g., 'the vent opening, which supports...').

When items in a list already contain commas, standard commas cannot be used to separate the main items, as this would cause confusion and create a run-on list.

3

Apply the appropriate punctuation rule for complex lists.

A semicolon must be used to separate the list items to maintain clarity.

Semicolons act as 'super-commas' to clearly demarcate where one complex list item ends and the next begins.

Key Concept

Semicolons in Complex Lists

Question 1552Question →

If $4^{x - 1} = 8$ , what is the value of $x$ ?

Show answer & explanation

Answer: 2.5

Answer

2.5 (or 5/2)

The correct answer is 2.5 (or 5/2). By expressing 4 as

2^2

and 8 as

2^3

, the equation is rewritten as

(2^2)^{x-1} = 2^3

. Applying the exponent power rule simplifies this to

2^{2x-2} = 2^3

. Equating the exponents yields the linear equation

2x-2 = 3

. Solving for

x

gives

2x = 5

, which results in

x = 2.5

or

5/2

.

Step-by-Step Solution

1

Rewrite both sides of the equation with a common base of 2.

(2^2)^{x-1} = 2^3

Expressing both bases as powers of 2 allows us to equate the exponents later.

2

Apply the power of a power exponent rule

(a^m)^n = a^{mn}

to the left side.

2^{2x-2} = 2^3

Multiplying the exponent 2 by the exponent

(x-1)

simplifies the expression to

2(x-1) = 2x-2

.

3

Equate the exponents and solve the linear equation for

x

.

x = 2.5

Since the bases are identical, their exponents must be equal, giving

2x-2 = 3

, which simplifies to

2x = 5

.

Key Concept

Solving exponential equations by finding a common base.

Question 1553Question →

The function $f$ is defined by $f(x) = x^2 - 10x + 29$ . For what value of $x$ does $f(x)$ reach its minimum value?

Show answer & explanation

Answer: 5

Answer

The function reaches its minimum value at

x = 5

.

A quadratic function defined by

f(x) = ax^2 + bx + c

where

a > 0

reaches its minimum value at its vertex. The

x

-coordinate of the vertex is calculated using the formula

x = -\frac{b}{2a}

. For

f(x) = x^2 - 10x + 29

, the coefficients are

a = 1

and

b = -10

. Substituting these values into the formula gives

x = -\frac{-10}{2(1)} = 5

.

Step-by-Step Solution

1

Identify the coefficients of the quadratic function in standard form

f(x) = ax^2 + bx + c

.

For

f(x) = x^2 - 10x + 29

, the coefficients are

a = 1

and

b = -10

.

To use the vertex formula

x = -\frac{b}{2a}

.

2

Apply the vertex formula to calculate the

x

-coordinate where the function reaches its minimum.

x = 5

.

Since the coefficient of

x^2

is positive (

a = 1

), the parabola opens upward, and its minimum value occurs at the vertex.

Key Concept

Finding the x-coordinate of the vertex of a quadratic function

Question 1554Question →

In the quadratic equation $x^2 - bx + 64 = 0$ , $b$ is a constant. The two real solutions to the equation are $r$ and $s$ , where $r = s^2$ . What is the value of $b$ ?

Show answer & explanation

Answer: 20

Answer

20

By applying Vieta's formulas to the quadratic equation

x^2 - bx + 64 = 0

, we find that the product of the solutions is

r \cdot s = 64

. Substituting the given relation

r = s^2

gives

s^3 = 64

, which yields the real solution

s = 4

. Substituting this back into the relation gives the other solution

r = 16

. Finally, the sum of the solutions is

r + s = b

, so

b = 16 + 4 = 20

.

Step-by-Step Solution

1

Apply Vieta's formula for the product of the roots

r \cdot s = 64

For a quadratic equation in the form

x^2 - bx + c = 0

, the product of the roots is equal to the constant term

c

.

2

Substitute the given root relationship into the product equation

s^3 = 64

, which solves to

s = 4

We are given that one root is the square of the other (

r = s^2

), so substituting

s^2

for

r

allows us to solve for

s

.

3

Calculate the value of the second root

r

r = 16

Using the relation

r = s^2

with

s = 4

, we find

r = 4^2 = 16

.

4

Apply Vieta's formula for the sum of the roots to find

b

b = 20

For the equation

x^2 - bx + 64 = 0

, the sum of the roots is

r + s = b

. Substituting

r = 16

and

s = 4

gives

16 + 4 = 20

.

Key Concept

Vieta's formulas relating the coefficients of a quadratic equation to its roots

Question 1555Question →

If $3^{2x - 1} = 27$ , what is the value of $x$ ?

1

5

2

14

Show answer & explanation

Answer: 2

Answer

2

To solve the equation

3^{2x - 1} = 27

, we first express 27 as a power of 3, which is

3^3

. Since the bases are the same, we set the exponents equal to each other to get

2x - 1 = 3

. Adding 1 to both sides gives

2x = 4

, and dividing both sides by 2 gives the value of

x

as 2.

Step-by-Step Solution

1

Express 27 with a base of 3

3^{2x - 1} = 3^3

To solve an exponential equation, we need to write both sides of the equation with a common base.

2

Set the exponents equal to each other

2x - 1 = 3

Since the bases are equal, their exponents must also be equal.

3

Solve the linear equation for

x

x = 2

Add 1 to both sides to get

2x = 4

, and then divide both sides by 2 to isolate

x

.

Key Concept

Solving exponential equations by finding a common base

Estimated Time:45s

Question 1556Question →

In the $xy$ -plane, the graph of $y = x^2$ is shifted $4$ units to the right and $9$ units up to produce the graph of $y = h(x)$ . Which of the following equations defines the function $h$ ?

h(x) = (x + 4)^2 + 9

h(x) = (x - 4)^2 - 9

h(x) = (x - 4)^2 + 9

h(x) = (x + 4)^2 - 9

Show answer & explanation

Answer: $h(x) = (x - 4)^2 + 9$

Answer

h(x) = (x - 4)^2 + 9

The parent function

y = x^2

has its vertex at

(0, 0)

. Shifting this graph

4

units to the right moves the x-coordinate of the vertex to

4

, and shifting it

9

units up moves the y-coordinate of the vertex to

9

. The new vertex is at

(4, 9)

. A parabola with a vertex of

(h, k)

and a leading coefficient of

1

is written in vertex form as

y = (x - h)^2 + k

. Substituting

h = 4

and

k = 9

gives the equation

h(x) = (x - 4)^2 + 9

.

Step-by-Step Solution

1

Identify the base function and the rules of graph translation.

The base function is

y = x^2

. Shifting a graph horizontally by

h

units changes the input

x

to

(x - h)

, and shifting vertically by

k

units adds

k

to the function value.

Applying transformations to the parent function changes its position in the coordinate plane while maintaining its shape.

2

Apply the horizontal translation of

4

units to the right.

Replacing

x

with

(x - 4)

in the parent function gives the intermediate equation

y = (x - 4)^2

.

A horizontal shift of

h

units to the right corresponds to replacing the input variable

x

with

(x - h)

.

3

Apply the vertical translation of

9

units up.

Adding

9

to the expression yields the final equation

h(x) = (x - 4)^2 + 9

.

A vertical shift of

k

units upward corresponds to adding

k

to the output of the function.

Key Concept

Quadratic graph transformations and translation rules

Question 1557Question →

If $4^x = 5$ , what is the value of $2^{4x - 1}$ ?

Show answer & explanation

Answer: 12.5

Answer

12.5

By writing

4^x

as

(2^2)^x = 2^{2x}

, we find that

2^{2x} = 5

. The expression

2^{4x-1}

can be rewritten using exponent properties as

\frac{2^{4x}}{2^1} = \frac{(2^{2x})^2}{2}

. Substituting

2^{2x} = 5

into this expression yields

\frac{5^2}{2} = \frac{25}{2} = 12.5

. Thus, the correct numerical response is 12.5.

Step-by-Step Solution

1

Express the given equation in terms of base 2.

2^{2x} = 5

Since

4 = 2^2

, we can rewrite

4^x

as

(2^2)^x

. Applying the power of a power rule,

(a^m)^n = a^{mn}

, gives

(2^2)^x = 2^{2x}

.

2

Rewrite the expression to be evaluated using exponent properties.

2^{4x - 1} = \frac{(2^{2x})^2}{2}

Using the division property of exponents,

a^{m-n} = \frac{a^m}{a^n}

, we can write

2^{4x - 1}

as

\frac{2^{4x}}{2}

. Then, using the power of a power rule in reverse,

2^{4x} = (2^{2x})^2

.

3

Substitute the known value of

2^{2x}

and simplify the numerical expression.

12.5

Substitute

2^{2x} = 5

into the expression

\frac{(2^{2x})^2}{2}

to obtain

\frac{5^2}{2} = \frac{25}{2}

, which simplifies to

12.5

.

Key Concept

Manipulating exponential equations by expressing bases in terms of common prime factors and applying properties of exponents.

Estimated Time:1m 30s

Question 1558Question →

The following passage is adapted from an essay about early twentieth-century astronomy. Which punctuation mark must be placed in the blank to properly connect the two independent clauses?

Fill in the blanks below

In the early twentieth century, Swedish astronomer Knut Lundmark was among the first to argue that some nebulae are actually distinct galaxies located far outside our own Milky Way. To support this claim, Lundmark analyzed the brightness of novae in the Andromeda Galaxy he concluded that Andromeda was millions of light-years away, a distance far greater than the size of our galaxy.

Show answer & explanation

Answer

A semicolon (;) is the correct punctuation mark to place in the blank to separate the two independent clauses.

A semicolon is correct because it links two closely related independent clauses without the need for a coordinating conjunction. In the passage, both the segment before the blank and the segment after the blank contain a subject and a verb and express a complete thought, meaning they are independent clauses.

Step-by-Step Solution

1

Analyze the grammatical structure of the clauses on either side of the blank.

The clause before the blank ('To support this claim, Lundmark analyzed the brightness of novae in the Andromeda Galaxy') and the clause after the blank ('he concluded that Andromeda was millions of light-years away') are both independent clauses capable of standing alone as complete sentences.

Identifying clause boundaries is necessary to determine standard punctuation usage.

2

Identify the relationship between the two independent clauses to select the appropriate punctuation mark.

The two clauses express closely related astronomical observations and are not joined by a coordinating conjunction. Therefore, they must be separated by a semicolon.

Standard English conventions require a semicolon (or a period) to link independent clauses that are not joined by a coordinating conjunction.

Key Concept

Using semicolons to connect independent clauses

Question 1559Question →

During the Devonian period, the evolution of deep-rooting vascular plants fundamentally altered the planet's geochemical systems. By fracturing bedrock, these early plants accelerated the weathering of silicate rocks, which absorbed vast amounts of atmospheric carbon dioxide ______ this carbon draw-down contributed to a global cooling trend that eventually triggered a series of major glaciation events.

Which choice completes the text so that it conforms to the conventions of Standard English?

dioxide, but

dioxide,

dioxide;

dioxide

Show answer & explanation

Answer: dioxide;

Answer

dioxide;

The sentence consists of two independent clauses: the first describing the absorption of atmospheric carbon dioxide, and the second explaining that this drawdown contributed to global cooling. Standard English conventions require two independent clauses to be joined by a semicolon, a period, or a comma with a coordinating conjunction. Using a semicolon correctly separates the clauses.

Step-by-Step Solution

1

Identify the boundary between the clauses and analyze their structure.

The text contains two distinct clauses: 'By fracturing bedrock, these early plants accelerated the weathering of silicate rocks, which absorbed vast amounts of atmospheric carbon dioxide' and 'this carbon draw-down contributed to a global cooling trend that eventually triggered a series of major glaciation events.'

Determining whether the clauses are independent or dependent dictates the required punctuation.

2

Determine if both clauses can stand independently.

Both clauses contain a subject and a verb and express a complete thought, making them independent clauses.

Independent clauses cannot be joined with just a comma or no punctuation; they require a semicolon, a period, or a comma and a coordinating conjunction.

3

Evaluate the choices based on standard punctuation rules and logical relationships.

The option containing a semicolon correctly links the two independent clauses. The option with a comma creates a comma splice, the option with no punctuation creates a run-on, and the option with 'but' introduces an illogical contrast.

A semicolon is the correct grammatical choice to connect closely related independent clauses without a conjunction.

Key Concept

Clause Boundaries and Linking

Estimated Time:1m 0s

Question 1560Question →

In the quadratic equation $x^2 - ax + b = 0$ , the roots $r$ and $s$ are positive integers. The quadratic equation $x^2 - (a+12)x + (b+12) = 0$ has roots $r^2$ and $s^2$ , where $a$ and $b$ are constants. What is the value of $a$ ?

Show answer & explanation

Answer: 5

Answer

The value of

a

is 5.

Applying Vieta's formulas to both equations and using the algebraic identity

r^2 + s^2 = (r+s)^2 - 2rs

yields the system of equations

a^2 - 2b = a + 12

and

b^2 = b + 12

. Solving the second equation gives

b = 4

(discarding the negative root since

r, s > 0

). Substituting

b = 4

into the first equation yields

a^2 - a - 20 = 0

, which solves to

a = 5

(discarding

a = -4

since

r, s > 0

).

Step-by-Step Solution

1

Apply Vieta's formulas to the first quadratic equation.

r+s = a

and

rs = b

To express the sum and product of the roots of the first equation in terms of the coefficients

a

and

b

.

2

Apply Vieta's formulas to the second quadratic equation.

r^2+s^2 = a+12

and

r^2s^2 = b+12

To express the sum and product of the squared roots in terms of the coefficients of the second equation.

3

Express

r^2+s^2

in terms of

a

and

b

using the algebraic expansion of a binomial square.

a^2 - 2b = a+12

By rewriting

r^2+s^2

as

(r+s)^2 - 2rs

, we can substitute

a

and

b

to form a system of equations.

4

Solve for the constant

b

using the product relation.

b = 4

The relation

(rs)^2 = b^2 = b+12

simplifies to the quadratic equation

b^2 - b - 12 = 0

. Factoring yields

b = 4

or

b = -3

. Since

r

and

s

are positive integers, their product

b

must be positive, meaning

b = 4

.

5

Substitute

b = 4

into the sum relation to solve for

a

.

a = 5

Substituting

b = 4

yields

a^2 - 2(4) = a+12

, which simplifies to

a^2 - a - 20 = 0

. Factoring yields

a = 5

or

a = -4

. Since

r

and

s

are positive integers, their sum

a

must be positive, meaning

a = 5

.

Key Concept

Vieta's Formulas and Symmetric Polynomial Relationships

Estimated Time:3m 0s

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