Tüm alıştırma soruları

Which of the following is equivalent to the expression $(4p - 1)(p + 2)$ ?

4p^2 + 7p - 2

4p^2 - 2

4p^2 + 9p - 2

11p - 2

Cevabı ve açıklamayı göster

Cevap: $4p^2 + 7p - 2$

Cevap

The equivalent expression is

4p^2 + 7p - 2

.

Using the distributive property (often remembered as FOIL), each term in the first binomial must be multiplied by each term in the second binomial. The products are

4p^2

,

8p

,

-p

, and

-2

. Combining the like terms (

8p

and

-p

) yields

7p

, making the final simplified expression

4p^2 + 7p - 2

.

Adım Adım Çözüm

1

Multiply the first terms of each binomial.

4p \times p = 4p^2

This establishes the quadratic term of the final polynomial.

2

Multiply the outer terms.

4p \times 2 = 8p

This finds the first part of the linear middle term.

3

Multiply the inner terms.

-1 \times p = -p

This finds the second part of the linear middle term.

4

Multiply the last terms.

-1 \times 2 = -2

This finds the constant term of the final polynomial.

5

Combine the like terms.

4p^2 + (8p - p) - 2 = 4p^2 + 7p - 2

Simplifying the expression converts it into standard polynomial form.

Anahtar Kavram

Multiplying Binomials

Soru 4982Soru →

Read the scenario below and solve for the unknown value. Enter your exact numerical answer in the blank provided.

Aşağıdaki boşlukları doldurun

A mechanic orders a set of spark plugs that cost

8 each, plus a flat shipping fee of

12. If the total charge is $60, the mechanic ordered spark plugs.

Cevabı ve açıklamayı göster

Cevap

6

By setting up the equation

8x + 12 = 60

, we can isolate

x

. Subtracting the constant

12

from both sides gives

8x = 48

. Dividing by

8

yields

x = 6

. Therefore,

6

spark plugs were ordered.

Adım Adım Çözüm

1

Set up a linear equation to represent the word problem.

8x + 12 = 60

Let

x

be the number of spark plugs. Multiplying the cost per plug (

8) by

x

and adding the flat shipping fee (

12) equals the total charge ($60).

2

Subtract 12 from both sides of the equation.

8x = 48

To begin isolating the variable

x

, the constant term must be removed from the left side of the equation.

3

Divide both sides by 8.

x = 6

To completely isolate

x

, divide by its coefficient. This reveals the number of spark plugs ordered.

Anahtar Kavram

Solving one-variable linear equations from a real-world scenario.

Soru 4983Soru →

A customer's monthly utility bill is calculated as twelve dollars less than three times the sum of their base usage fee, $f$ , and five. If the total bill amounts to $108$ , which of the following equations correctly models this calculation?

3(f + 5) - 12 = 108

12 - 3(f + 5) = 108

3f + 5 - 12 = 108

3(5f) - 12 = 108

Cevabı ve açıklamayı göster

Cevap: $3(f + 5) - 12 = 108$

Cevap

The equation that correctly represents the calculation is

3(f + 5) - 12 = 108

.

The equation is built step-by-step: 'the sum of their base usage fee,

f

, and five' is written as

(f + 5)

. 'Three times' this sum requires multiplying the grouped expression, resulting in

3(f + 5)

. Finally, 'twelve dollars less than' means

12

must be subtracted from that product, yielding the final correct equation

3(f + 5) - 12 = 108

.

Adım Adım Çözüm

1

Translate the phrase 'the sum of their base usage fee,

f

, and five'.

(f + 5)

The phrase 'the sum of' indicates addition between the variable

f

and the number

5

. Parentheses are required because a subsequent operation applies to this entire sum.

2

Translate 'three times the sum'.

3(f + 5)

The multiplier

3

must distribute across the entire sum, which is represented by placing it directly outside the parentheses.

3

Translate 'twelve dollars less than'.

3(f + 5) - 12

The phrase 'less than' is a turnaround phrase meaning

12

is subtracted from the quantity that precedes it conceptually, placing

- 12

at the end.

4

Form the complete equation by setting the expression equal to the total bill.

3(f + 5) - 12 = 108

The phrase 'amounts to' signifies equality with the final given value of

108

.

Anahtar Kavram

Translating complex word phrases into algebraic equations requires careful attention to operation order, turnaround words like 'less than', and grouping symbols for terms like 'the sum of'.

Tahmini Süre:1m 0s

Soru 4984Soru →

A communications technician is configuring signal transponders. The synchronization cycle for each transponder is determined by the Least Common Multiple (LCM) of its two input frequencies (measured in Hz). Arrange the following pairs of input frequencies in ascending order based on their resulting synchronization cycle (smallest LCM to largest LCM).

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct order from smallest to largest LCM is: 12 Hz and 18 Hz (LCM=36), 16 Hz and 24 Hz (LCM=48), 10 Hz and 25 Hz (LCM=50), 8 Hz and 14 Hz (LCM=56), followed by 15 Hz and 20 Hz (LCM=60).

The correct sequence is determined by finding the LCM of each frequency pair and listing them in ascending order. The calculated LCMs are 36, 48, 50, 56, and 60 respectively.

Adım Adım Çözüm

1

Calculate the Least Common Multiple (LCM) for the first pair,

12\text{ Hz}

and

18\text{ Hz}

.

The LCM of

12

and

18

is

36

.

The multiples of

18

are

18, 36, 54

. Since

36

is divisible by

12

(

12 \times 3 = 36

), it is the lowest common multiple.

2

Calculate the LCM for

16\text{ Hz}

and

24\text{ Hz}

.

The LCM of

16

and

24

is

48

.

The multiples of

24

are

24, 48

. Since

48

is divisible by

16

(

16 \times 3 = 48

), it is the lowest common multiple.

3

Calculate the LCM for

10\text{ Hz}

and

25\text{ Hz}

.

The LCM of

10

and

25

is

50

.

The multiples of

25

are

25, 50

. Since

50

is divisible by

10

(

10 \times 5 = 50

), it is the lowest common multiple.

4

Calculate the LCM for

8\text{ Hz}

and

14\text{ Hz}

.

The LCM of

8

and

14

is

56

.

Using prime factorizations:

8 = 2^3

and

14 = 2 \times 7

. The LCM takes the highest powers:

2^3 \times 7 = 8 \times 7 = 56

.

5

Calculate the LCM for

15\text{ Hz}

and

20\text{ Hz}

.

The LCM of

15

and

20

is

60

.

Using prime factorizations:

15 = 3 \times 5

and

20 = 2^2 \times 5

. The LCM takes the highest powers:

2^2 \times 3 \times 5 = 60

.

6

Arrange the pairs in ascending order based on their calculated LCMs.

36 < 48 < 50 < 56 < 60

The stem requires ordering the frequencies from the smallest resulting synchronization cycle (smallest LCM) to the largest.

Anahtar Kavram

Calculating the Least Common Multiple (LCM) to synchronize multiple periodic events.

Soru 4985Soru →

An experimental particle travels a distance of $4.5 \times 10^5$ meters in $5.0 \times 10^{-3}$ seconds. What is the average speed of the particle in meters per second, expressed in proper scientific notation?

9.0 \times 10^7

9.0 \times 10^1

9.0 \times 10^9

9.0 \times 10^8

Cevabı ve açıklamayı göster

Cevap: $9.0 \times 10^7$

Cevap

The correct speed is

9.0 \times 10^7

meters per second.

To divide numbers in scientific notation, divide the coefficients and subtract the exponents. Dividing

4.5

by

5.0

yields

0.9

. Subtracting the exponent

-3

from

5

yields

8

(since

5 - (-3) = 5 + 3 = 8

). This gives an intermediate result of

0.9 \times 10^8

. To convert this into proper scientific notation, the coefficient must be between

1

and

10

. Moving the decimal point one place to the right converts

0.9

to

9.0

, which requires decreasing the exponent by

1

, resulting in the final correct answer of

9.0 \times 10^7

.

Adım Adım Çözüm

1

Divide the coefficients given in the problem:

4.5 \div 5.0

.

0.9

When dividing numbers in scientific notation, the coefficients are divided separately from the powers of 10.

2

Divide the powers of 10 by subtracting the denominator's exponent from the numerator's exponent:

10^5 \div 10^{-3}

.

10^{5 - (-3)} = 10^8

According to the laws of exponents, division of like bases requires subtracting the exponent of the divisor from the exponent of the dividend.

3

Combine the newly calculated coefficient and power of 10.

0.9 \times 10^8

This represents the mathematical result before applying proper formatting rules.

4

Normalize the answer by moving the decimal point one place to the right to make the coefficient between 1 and 10, and subtract 1 from the exponent to balance the shift.

9.0 \times 10^7

Proper scientific notation requires the coefficient to be greater than or equal to 1 and less than 10.

Anahtar Kavram

Scientific Notation Division and Normalization

Soru 4986Soru →

A logistics coordinator is packing identical emergency relief kits. They have $180$ bottles of water, $252$ energy bars, and $324$ bandages. If every kit must contain the exact same combination of supplies with no items left over, what is the greatest number of kits that can be assembled?

18

36

72

11,340

Cevabı ve açıklamayı göster

Cevap: 36

Cevap

The greatest number of kits that can be assembled is 36.

The scenario describes dividing three sets of items into identical groups with no remainder, which means finding the Greatest Common Factor (GCF). Breaking down the numbers into prime factors yields:

180 = 2^2 \times 3^2 \times 5

,

252 = 2^2 \times 3^2 \times 7

, and

324 = 2^2 \times 3^4

. The factors shared by all three are

2^2

and

3^2

. Multiplying these together (

4 \times 9

) gives the GCF of

36

. Therefore, a maximum of

36

kits can be made, each containing exactly

5

water bottles,

7

energy bars, and

9

bandages.

Adım Adım Çözüm

1

Identify the mathematical operation required by the scenario.

We need to find the Greatest Common Factor (GCF) of 180, 252, and 324.

The problem asks for the maximum number of identical kits that can be made by dividing all supplies without any leftovers, which aligns with the definition of the GCF.

2

Determine the prime factorization for each quantity.

180 = 2^2 \times 3^2 \times 5

252 = 2^2 \times 3^2 \times 7

324 = 2^2 \times 3^4

Prime factorization is a reliable method to find all common prime factors among multiple large numbers.

3

Identify the common prime factors and their lowest exponents.

The common prime bases are 2 and 3. Their lowest exponents across all factorizations are

2^2

and

3^2

.

The GCF is the product of the lowest power of each prime factor that appears in all of the numbers.

4

Calculate the final GCF by multiplying the common prime factors.

2^2 \times 3^2 = 4 \times 9 = 36

Multiplying the lowest common prime factors gives the greatest numerical value that divides evenly into all three supply quantities.

Anahtar Kavram

Greatest Common Factor (GCF) via Prime Factorization

Alternatif Yöntem

Instead of prime factorization, you can check the factors of the common difference. The difference between the quantities is

72

(

252 - 180 = 72

and

324 - 252 = 72

). The factors of

72

are

72, 36, 24, 18, 12, 9, 8, 6, 4, 3, 2, 1

. Checking these against

180

from largest to smallest,

72

does not divide

180

, but

36

does (

180 \div 36 = 5

). Since

36

divides all three numbers, it is the GCF.

Tahmini Süre:1m 30s

Soru 4987Soru →

Match each verbal phrase describing a mathematical relationship or scenario to its correct algebraic expression.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The correct pairings match each contextual or mathematical description with its exact symbolic translation, respecting order of operations, geometric formulas, and grouping words.

Each correct match successfully identifies key translation words ('product', 'quotient', 'less than', 'quantity') and applies proper algebraic syntax, including necessary parentheses, geometric formulas, and the correct order for subtraction.

Adım Adım Çözüm

1

Translate the rate and discount scenario.

The phrase '

k

items at twelve dollars each' translates to

12k

. The phrase 'minus a five-dollar discount' makes it

12k - 5

.

Multiplication represents a per-item rate, and a discount implies subtraction.

2

Translate the quotient phrase.

The 'quotient of

k

and five' is

\frac{k}{5}

. 'Twelve less than' means subtracting

12

from this term, yielding

\frac{k}{5} - 12

.

The phrase 'less than' dictates reversing the order of the terms in the algebraic subtraction.

3

Translate the grouped product phrase.

The 'quantity of five less than

k

' is translated as

(k - 5)

. The 'product of twelve and the quantity' is

12(k - 5)

.

The word 'quantity' signals that the addition or subtraction must be grouped in parentheses before applying the multiplication.

4

Translate the geometric scenario.

The 'area of a rectangle with length twelve and width

k

' translates to

12k

. 'Increased by five' adds

5

, yielding

12k + 5

.

Area of a rectangle is calculated by multiplying length and width, and 'increased by' signifies addition.

Anahtar Kavram

Translating Word Phrases into Algebraic Equations

Soru 4988Soru →

Match each unsimplified algebraic expression on the left with its equivalent, fully simplified form on the right. Assume all variables represent non-zero real numbers.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Cevabı ve açıklamayı göster

Cevap

The expressions match as follows: (3a^4b^2)(4a^2b) is equivalent to 12a^6b^3. The quotient (24a^7b^3)/(6a^3b^6) simplifies to 4a^4b^-3. The power (2a^3b^-2)^4 expands to 16a^12b^-8. Finally, (5a^0b^3)^-2 simplifies to 1/(25b^6).

Each expression simplifies directly using the fundamental rules of exponents: product rule (add exponents), quotient rule (subtract exponents), power rule (multiply exponents), and zero/negative exponent rules.

Adım Adım Çözüm

1

Simplify the first expression by multiplying the coefficients and applying the product rule for exponents.

3 * 4 = 12, a^(4+2) = a^6, b^(2+1) = b^3. Combined: 12a^6b^3

When multiplying terms with the same base, add their exponents.

2

Simplify the second expression by dividing the coefficients and applying the quotient rule for exponents.

24 / 6 = 4, a^(7-3) = a^4, b^(3-6) = b^-3. Combined: 4a^4b^-3

When dividing terms with the same base, subtract the denominator's exponent from the numerator's exponent.

3

Simplify the third expression by applying the power of a product rule.

2^4 = 16, a^(3*4) = a^12, b^(-2*4) = b^-8. Combined: 16a^12b^-8

When raising a power to a power, multiply the exponents. Apply the exponent to the numerical coefficient as well.

4

Simplify the fourth expression by applying the zero exponent rule, then the power rule.

a^0 = 1. 5^-2 = 1/25. b^(3*-2) = b^-6. Combined: 1/(25b^6)

Any non-zero term raised to the power of 0 is 1. Negative exponents indicate taking the reciprocal.

Anahtar Kavram

Applying the product, quotient, power, zero, and negative rules of exponents to simplify algebraic expressions with multiple variables and coefficients.

Soru 4989Soru →

What value of

m

makes the equation

12m - 7 = 41

true?

Cevabı ve açıklamayı göster

Cevap: 4

Cevap

4

Applying the correct sequence of inverse operations (first adding 7, then dividing by 12) yields exactly 4.

Adım Adım Çözüm

1

Add 7 to both sides of the equation.

12m = 48

This is the first inverse operation needed to isolate the variable term.

2

Divide both sides by 12.

m = 4

This inverse operation isolates

m

, giving the final solution.

Anahtar Kavram

Solving two-step linear equations using inverse operations

Tahmini Süre:30s

Soru 4990Soru →

Consider the algebraic expression below, where $x$ and $y$ represent non-zero values:

\frac{(4x^3 y^{-2})^2}{2x^{-1} y^3}

Which of the following represents the completely simplified form of this expression?

\frac{8x^5}{y^7}

\frac{8x^7}{y^7}

\frac{2x^7}{y^7}

\frac{8x^6}{y^3}

Cevabı ve açıklamayı göster

Cevap: $\frac{8x^7}{y^7}$

Cevap

The completely simplified form is

\frac{8x^7}{y^7}

.

The expression is simplified by first distributing the outer exponent of 2 to every factor in the numerator, which squares the coefficient 4 and multiplies the variables' exponents to produce

16x^6y^{-4}

. Next, dividing the coefficients yields

16 \div 2 = 8

. Then, the quotient rule is applied by subtracting the denominator's exponents from the numerator's exponents for matching bases: for

x

,

6 - (-1) = 7

, and for

y

,

-4 - 3 = -7

. This results in

8x^7y^{-7}

. Finally, the negative exponent is resolved by moving

y^7

to the denominator, giving the completely simplified form

\frac{8x^7}{y^7}

.

Adım Adım Çözüm

1

Apply the power of a product rule to simplify the numerator.

(4x^3 y^{-2})^2 = 4^2 \cdot (x^3)^2 \cdot (y^{-2})^2 = 16x^6y^{-4}

When an algebraic term is raised to a power, the outer exponent multiplies the exponent of each factor inside the parentheses, including the numerical coefficient (which has an implied exponent of 1).

2

Rewrite the fraction using the expanded numerator.

\frac{16x^6y^{-4}}{2x^{-1} y^3}

This sets up the expression to cleanly apply the quotient rules for exponents and to divide the numerical coefficients.

3

Divide the numerical coefficients.

16 \div 2 = 8

Coefficients follow standard arithmetic division rules independent of the variables.

4

Apply the quotient rule to the variables by subtracting the denominator's exponents from the numerator's exponents.

For

x

:

6 - (-1) = 6 + 1 = 7

. For

y

:

-4 - 3 = -7

.

The quotient rule states that

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

. Note that subtracting a negative number is algebraically equivalent to addition.

5

Combine the terms and convert the negative exponent into a positive exponent.

8x^7y^{-7} = \frac{8x^7}{y^7}

A negative exponent represents the reciprocal of the base (

a^{-n} = \frac{1}{a^n}

). Moving

y^{-7}

to the denominator as

y^7

yields the final simplified expression format.

Anahtar Kavram

Operations with Variable Exponents

Soru 4991Soru →

Determine the missing coefficient in the area expression.

Aşağıdaki boşlukları doldurun

A rectangle has a length of

(3x - 4)

units and a width of

(x + 6)

units. The area of the rectangle can be represented by the expanded polynomial $3x^2 + x - 24$ square units.

Cevabı ve açıklamayı göster

Cevap

14

The area of the rectangle is calculated by multiplying its length and width:

(3x - 4)(x + 6)

. Distributing the terms, the outer product is

18x

and the inner product is

-4x

. Combining these like terms (

18x - 4x

) results in

14x

. Therefore, the missing coefficient for the

x

term is 14.

Adım Adım Çözüm

1

Set up the multiplication for the area of the rectangle.

(3x - 4)(x + 6)

The area of a rectangle is found by multiplying its length and its width.

2

Multiply the binomials using the distributive property.

3x^2 + 18x - 4x - 24

Each term in the first binomial must be multiplied by each term in the second binomial.

3

Combine the like terms containing the

x

variable.

18x - 4x = 14x

Simplifying the expression requires combining terms that share the same variable and exponent.

Anahtar Kavram

Multiplying binomials and combining like terms

Soru 4992Soru →

A tactical transport vehicle's fuel efficiency is evaluated under two conditions. On paved roads, the ratio of miles driven to gallons of fuel consumed is $15:1$ . In off-road conditions, the vehicle's fuel efficiency decreases by $20\%$ . How many gallons of fuel are required to drive $60\text{ miles}$ in off-road conditions?

4.8

3.2

5

0.2

Cevabı ve açıklamayı göster

Cevap: $5$

Cevap

The correct amount of fuel required is

5

gallons.

The original fuel efficiency is 15 miles per gallon. A 20% decrease reduces this rate by 3 mpg (0.20 × 15), resulting in a new off-road efficiency of 12 mpg. To find the required fuel for a 60-mile trip, divide the total distance by the new unit rate: 60 / 12 = 5 gallons.

Adım Adım Çözüm

1

Identify the original fuel efficiency from the given ratio.

The ratio

15:1

means the vehicle gets

15\text{ miles per gallon}

(mpg) on paved roads.

Converting the ratio into a unit rate makes it easier to apply the percentage change.

2

Calculate the off-road fuel efficiency by applying the

20\%

decrease.

15 - (0.20 \times 15) = 15 - 3 = 12\text{ mpg}

.

The problem states the efficiency decreases by

20\%

in off-road conditions.

3

Set up a proportion to find the gallons required for the

60\text{-mile}

trip.

\frac{12\text{ miles}}{1\text{ gallon}} = \frac{60\text{ miles}}{x\text{ gallons}}

Fuel consumption is a direct proportion between miles driven and gallons used at a constant rate.

4

Solve for

x

.

12x = 60 \implies x = 5\text{ gallons}

.

Isolating the variable gives the final required amount of fuel.

Anahtar Kavram

Applying percentage changes to unit rates and setting up proportions to solve for an unknown quantity.

Alternatif Yöntem

Calculate the fuel needed for paved roads first (

60 / 15 = 4\text{ gallons}

). Recognize that since the new efficiency is

80\%

(or

\frac{4}{5}

) of the original, the fuel needed will be the inverse ratio (

\frac{5}{4}

) of the original fuel amount. Multiply

4\text{ gallons} \times \frac{5}{4} = 5\text{ gallons}

.

Tahmini Süre:1m 0s

Soru 4993Soru →

Calculate the product of the binomials $(m - 6)$ and $(m + 2)$ . What is the resulting polynomial in simplest form?

m^2 - 4m - 12

m^2 - 12

m^2 + 4m - 12

m^2 - 4m + 12

Cevabı ve açıklamayı göster

Cevap: $m^2 - 4m - 12$

Cevap

The correct answer is the polynomial

m^2 - 4m - 12

.

The correct answer is obtained by distributing each term in the first binomial to each term in the second binomial, which gives

m^2 + 2m - 6m - 12

. Combining the like terms (

2m - 6m

) yields

-4m

, resulting in the final simplified polynomial

m^2 - 4m - 12

.

Adım Adım Çözüm

1

Apply the distributive property to multiply each term in the first binomial by each term in the second binomial.

m \cdot m + m \cdot 2 - 6 \cdot m - 6 \cdot 2

To expand the product of the two binomial expressions.

2

Perform the individual multiplications for each term.

m^2 + 2m - 6m - 12

To simplify the expression before combining any like terms.

3

Combine the like terms (the middle terms containing

m

).

m^2 - 4m - 12

To write the polynomial in its standard, fully simplified form.

Anahtar Kavram

Multiplying Polynomials

Alternatif Yöntem

You can use the area model (or box method) to organize the multiplication. Draw a 2x2 grid, place the terms

m

and

-6

along the top edge, and

m

and

2

along the side. Multiply the intersecting terms to fill the four cells (

m^2

,

-6m

,

2m

, and

-12

), and then sum all the cells to find the final polynomial.

Tahmini Süre:45s

Soru 4994Soru →

A rectangular microchip has a length of $8.0 \times 10^{-3}$ meters and a width of $4.5 \times 10^{-3}$ meters. Calculate the area of the microchip in square meters and express the result in proper scientific notation as $a \times 10^n$ (where $1 \le a < 10$ ). What is the value of the coefficient $a$ ?

Cevabı ve açıklamayı göster

Cevap: 3.6

Cevap

The coefficient

a

is 3.6.

The correct answer is derived by multiplying the coefficients (8.0 × 4.5 = 36), adding the exponents (-3 + -3 = -6), and normalizing the preliminary result of 36 × 10^-6 by moving the decimal left by one place. This creates a proper coefficient of 3.6 and changes the exponent to -5.

Adım Adım Çözüm

1

Set up the multiplication for area

Area =

(8.0 \times 10^{-3}) \times (4.5 \times 10^{-3})

The area of a rectangle is found by multiplying its length by its width.

2

Multiply the coefficients

8.0 \times 4.5 = 36

When multiplying terms in scientific notation, the numerical coefficients are multiplied together.

3

Multiply the powers of 10

10^{-3} \times 10^{-3} = 10^{-6}

When multiplying identical bases, their exponents are added:

-3 + (-3) = -6

.

4

Combine into preliminary result

36 \times 10^{-6}

Bring the multiplied coefficient and combined power of 10 together.

5

Normalize to proper scientific notation

3.6 \times 10^{-5}

Proper scientific notation requires the coefficient

a

to satisfy

1 \le a < 10

. Moving the decimal one place left on 36 increases the exponent by 1 (

-6 + 1 = -5

).

6

Extract the coefficient

a

3.6

The question specifically asks for the value of the coefficient

a

in the final

a \times 10^n

format.

Anahtar Kavram

Multiplying and Normalizing Values in Scientific Notation

Soru 4995Soru →

The expression $(2y - 7)^2$ is expanded into the standard polynomial form $ay^2 + by + c$ . What is the value of the coefficient $b$ ?

Cevabı ve açıklamayı göster

Cevap: -28

Cevap

-28

Expanding the perfect square binomial

(2y - 7)^2

requires multiplying

(2y - 7)

by itself. Using the distributive property, the outer product is

(2y)(-7) = -14y

and the inner product is

(-7)(2y) = -14y

. Combining these like terms gives a total middle term of

-28y

. Because the required polynomial form is

ay^2 + by + c

, the value of the coefficient

b

is exactly -28.

Adım Adım Çözüm

1

Rewrite the expression as a product of two binomials.

(2y - 7)(2y - 7)

Writing the binomial twice makes it easier to correctly apply the distributive property (FOIL method) and avoid missing the middle terms.

2

Distribute the terms.

4y^2 - 14y - 14y + 49

Multiplying the First terms, Outer terms, Inner terms, and Last terms ensures that every term in the first binomial is multiplied by every term in the second binomial.

3

Combine the like terms.

4y^2 - 28y + 49

The middle terms,

-14y

and

-14y

, have the same variable raised to the same power, so their coefficients can be added together.

4

Identify the target coefficient.

-28

The standard polynomial form is given as

ay^2 + by + c

. The value of

b

corresponds to the coefficient of the

y

term.

Anahtar Kavram

Expanding Perfect Square Binomials

Soru 4996Soru →

A plumber has a copper pipe that is $14.5$ inches long. She cuts off a piece that is $3.75$ inches long. What is the length of the remaining pipe in inches?

Cevabı ve açıklamayı göster

Cevap: 10.75

Cevap

10.75

Subtracting the cut piece (

3.75

inches) from the total length (

14.5

inches) gives the remaining length. Aligning the decimals properly as

14.50 - 3.75

results in exactly

10.75

.

Adım Adım Çözüm

1

Identify the operation needed

Subtraction:

14.5 - 3.75

Cutting a piece off means reducing the total length of the pipe.

2

Align the decimals for subtraction

14.50 - 3.75

To subtract decimals correctly, align the place values by lining up the decimal points. Add a placeholder zero to

14.5

to make it

14.50

.

3

Perform the subtraction

10.75

Subtracting

3.75

from

14.50

yields

10.75

.

Anahtar Kavram

Decimal Subtraction and Alignment

Tahmini Süre:45s

Soru 4997Soru →

Given the expression $7x - 2y^2$ , what is its numerical value when $x = 5$ and $y = 3$ ?

Cevabı ve açıklamayı göster

Cevap: 17

Cevap

17

The expression evaluates to 17 by successfully substituting the values into the variables and properly applying the order of operations.

Adım Adım Çözüm

1

Substitute the values

x = 5

and

y = 3

into the algebraic expression.

7(5) - 2(3)^2

Variables must be replaced with their numerical values before evaluating.

2

Calculate the exponent.

7(5) - 2(9)

According to the order of operations (PEMDAS), exponents must be evaluated before multiplication.

3

Perform the multiplication operations.

35 - 18

Multiplication is evaluated before subtraction.

4

Subtract the resulting numbers.

17

This final arithmetic step yields the evaluated expression.

Anahtar Kavram

Evaluating Algebraic Expressions

Soru 4998Soru →

Consider the linear equation below:

6y - 18 = 24

What is the value of $y$ ?

7

1

-2

22

Cevabı ve açıklamayı göster

Cevap: 7

Cevap

The correct value is 7.

The correct answer is found by first adding 18 to both sides of the equation to get

6y = 42

. Then, dividing both sides by 6 isolates

y

, resulting in

y = 7

.

Adım Adım Çözüm

1

Isolate the term with the variable by adding 18 to both sides of the equation.

6y = 42

Applying the addition property of equality cancels out the -18 on the left side.

2

Solve for the variable by dividing both sides by 6.

y = 7

Dividing by the coefficient of the variable isolates

y

and provides the final solution.

Anahtar Kavram

Solving linear equations by isolating the variable using inverse operations.

Soru 4999Soru →

Determine the final value of each arithmetic expression below. What is the correct sequence when the expressions are arranged from smallest to largest result?

Öğeleri doğru sıraya koymak için sürükleyin

Cevabı ve açıklamayı göster

Cevap

The correct sequence from smallest to largest is:

0.18 + 0.42

,

3.5 \div 5

,

2.1 - 1.3

, and

0.15 \times 6

.

To arrange the sequence properly, first calculate their values:

0.18 + 0.42 = 0.6

,

3.5 \div 5 = 0.7

,

2.1 - 1.3 = 0.8

, and

0.15 \times 6 = 0.9

. By comparing these results side by side, the accurate ascending order is

0.6

,

0.7

,

0.8

, and

0.9

.

Adım Adım Çözüm

1

Calculate the sum for the fourth expression.

0.18 + 0.42 = 0.60

, which is

0.6

.

Adding the hundredths and tenths places properly yields

0.6

.

2

Calculate the quotient for the third expression.

3.5 \div 5 = 0.7

.

Since

35 \div 5 = 7

, dividing

3.5

by

5

gives

0.7

.

3

Calculate the difference for the first expression.

2.1 - 1.3 = 0.8

.

Borrowing from the ones place allows subtracting

1.3

from

2.1

to get

0.8

.

4

Calculate the product for the second expression.

0.15 \times 6 = 0.90

, which is

0.9

.

Multiplying

15

by

6

gives

90

. Since the original factors have a total of two decimal places, the result is

0.90

.

5

Order the evaluated results from smallest to largest.

0.6 < 0.7 < 0.8 < 0.9

.

Comparing the tenths digit of each calculated value provides the final arrangement.

Anahtar Kavram

Performing basic arithmetic operations with decimals and comparing their magnitudes.

Soru 5000Soru →

The maximum payload capacity, $C$ , of a transport drone is equal to twice the sum of its base weight, $w$ , and fifty, subtracted from five hundred. Which of the following equations correctly models this capacity?

C = 500 - 2(w + 50)

C = 2(w + 50) - 500

C = 500 - 2w + 50

C = 2w + 50 - 500

Cevabı ve açıklamayı göster

Cevap: $C = 500 - 2(w + 50)$

Cevap

The correct equation is

C = 500 - 2(w + 50)

.

The correct equation translates the text sequentially while respecting mathematical grouping. First, 'the sum of its base weight,

w

, and fifty' translates to the grouped expression

(w + 50)

. 'Twice the sum' means multiplying that entire group by 2, yielding

2(w + 50)

. Finally, 'subtracted from five hundred' dictates the order of the subtraction: 500 is the initial amount from which the expression is taken away, resulting in

500 - 2(w + 50)

.

Adım Adım Çözüm

1

Translate the phrase 'the sum of its base weight, w, and fifty'.

(w + 50)

The word 'sum' indicates addition, and since it is a single combined quantity, it must be enclosed in parentheses.

2

Translate the phrase 'twice the sum'.

2(w + 50)

The word 'twice' means multiplying by 2. It applies to the entire sum.

3

Translate the phrase 'subtracted from five hundred' to form the final equation.

C = 500 - 2(w + 50)

The phrase 'subtracted from X' means X is the starting amount. Therefore, the expression

2(w + 50)

must be subtracted from 500.

Anahtar Kavram

Translating Word Phrases into Algebraic Equations

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar