Examkin

Algebra

282 questions

All types282Multiple choice147True / False7Numeric33Matching33Fill in the blank39Short answer23
Question 201Question →

What is the coefficient of the kk term in the expanded product of (3k−2)(3k - 2) and (k+5)(k + 5)?

Show answer & explanation

Answer: 13

Answer

13
When expanding (3k−2)(k+5)(3k - 2)(k + 5), the kk terms are generated by multiplying the outer terms (3k×5=15k)(3k \times 5 = 15k) and the inner terms (−2×k=−2k)(-2 \times k = -2k). Combining these gives 15k−2k=13k15k - 2k = 13k. Therefore, the coefficient of the kk term is 13.

Step-by-Step Solution

1
Identify the terms that will produce a kk term when multiplied.
The outer terms are 3k3k and 55. The inner terms are −2-2 and kk.
Only the product of a variable term and a constant term will result in a term with degree 1 (kk).
2
Multiply the outer terms.
(3k)(5)=15k(3k)(5) = 15k
This is the 'Outer' multiplication step in expanding two binomials.
3
Multiply the inner terms.
(−2)(k)=−2k(-2)(k) = -2k
This is the 'Inner' multiplication step in expanding two binomials.
4
Combine the resulting like terms.
15k+(−2k)=13k15k + (-2k) = 13k
Combining like terms gives the final kk term of the expanded polynomial.
5
Identify the coefficient.
13
The coefficient is the numerical factor of the kk term.

Key Concept

Multiplying polynomials and combining like terms
Question 202Question →

If x=−3x = -3 and y=−4y = -4, what does the expression 5x−2y5x - 2y equal?

Show answer & explanation

Answer: -7

Answer

-7
Substituting the given values into the expression yields 5(−3)−2(−4)5(-3) - 2(-4). Multiplying the terms results in −15−(−8)-15 - (-8), which simplifies to −15+8-15 + 8. Adding these numbers correctly produces −7-7.

Step-by-Step Solution

1
Substitute x=−3x = -3 and y=−4y = -4 into the algebraic expression.
5(−3)−2(−4)5(-3) - 2(-4)
Variable substitution replaces each variable with its given numerical value using parentheses.
2
Multiply the terms according to the order of operations.
−15−(−8)-15 - (-8)
Multiplication is performed before subtraction: 5×−3=−155 \times -3 = -15 and 2×−4=−82 \times -4 = -8.
3
Simplify the double negative and add the integers.
−15+8=−7-15 + 8 = -7
Subtracting a negative number is equivalent to adding its positive counterpart.

Key Concept

Evaluating Algebraic Expressions
Estimated Time:45s
Question 203Question →
Find the solution to the following algebraic equation:
4(3r−6)=48 4(3r - 6) = 48
Which of the following is the correct value of rr?
Show answer & explanation

Answer: 6

Answer

The correct value is 6.
The correct answer is found by first distributing the 4 to get 12r - 24 = 48. Next, adding 24 to both sides gives 12r = 72. Finally, dividing both sides by 12 yields r = 6.

Step-by-Step Solution

1
Distribute the 4 into the terms inside the parentheses.
12r−24=48 12r - 24 = 48
To eliminate the parentheses and simplify the left side of the equation.
2
Add 24 to both sides of the equation.
12r=72 12r = 72
To isolate the term containing the variable rr on one side of the equation using inverse operations.
3
Divide both sides of the equation by 12.
r=6 r = 6
To solve for the final, single value of the variable rr.

Key Concept

Solving One-Variable Linear Equations
Question 204Question →

A structural engineer calculates the maximum safe load, LL, for a support beam. The safe load is equal to fifty less than triple the sum of the beam's cross-sectional area, AA, and twelve. What algebraic equation models the maximum safe load?

Show answer & explanation

Answer: L=3(A+12)−50L = 3(A + 12) - 50

Answer

The correct equation is L=3(A+12)−50L = 3(A + 12) - 50.
The correct equation methodically translates the text: 'the sum of... A, and twelve' becomes the grouped quantity (A + 12). 'Triple' multiplies this entire group by 3, giving 3(A + 12). Finally, 'fifty less than' dictates subtracting 50 from that product, yielding 3(A + 12) - 50.

Step-by-Step Solution

1
Identify the base quantity being modified.
The phrase 'the sum of the beam's cross-sectional area, AA, and twelve' translates algebraically to the grouped expression (A+12)(A + 12).
The word 'sum' implies addition, and it creates a single quantity that must be grouped in parentheses before further operations are applied.
2
Translate the multiplier.
The word 'triple' indicates multiplication by 3, resulting in the expression 3(A+12)3(A + 12).
The multiplier applies to the entire sum identified in the previous step.
3
Translate the subtraction phrase.
The phrase 'fifty less than' means subtracting 50 from the previously established quantity, yielding 3(A+12)−503(A + 12) - 50.
In algebraic translation, 'X less than Y' dictates an order reversal to become Y - X.
4
Form the final equation.
Set the maximum safe load, LL, equal to the derived expression: L=3(A+12)−50L = 3(A + 12) - 50.
The problem states that the safe load is equal to the verbal expression.

Key Concept

Translating order-dependent phrases and implied grouping symbols from English into algebraic notation.
Estimated Time:45s
Question 205Question →

In a structural modeling calculation, an engineer must simplify a given formula before evaluating it. Which of the following expressions is mathematically equivalent to the algebraic fraction below?

16a5b2(2ab−1)3\frac{16a^5b^2}{(2ab^{-1})^3}
Show answer & explanation

Answer: 2a2b52a^2b^5

Answer

The mathematically equivalent expression is 2a2b52a^2b^5.
The correct answer is derived by first applying the power rule to the entire denominator, yielding 8a3b−38a^3b^{-3}. Next, dividing the numerical coefficients (16÷8=216 \div 8 = 2) and applying the quotient rule for exponents (subtracting the denominator's exponents from the numerator's exponents) results in a5−3=a2a^{5-3} = a^2 and b2−(−3)=b2+3=b5b^{2-(-3)} = b^{2+3} = b^5. Combining these distinct parts yields 2a2b52a^2b^5.

Step-by-Step Solution

1
Apply the power of a product rule to the denominator.
(2ab−1)3=23⋅a3⋅(b−1)3(2ab^{-1})^3 = 2^3 \cdot a^3 \cdot (b^{-1})^3
An exponent outside parentheses must be applied to every distinct factor inside the parentheses.
2
Simplify the factors in the denominator.
8a3b−38a^3b^{-3}
23=82^3 = 8, and for a power of a power (b−1)3(b^{-1})^3, the exponents are multiplied (−1×3=−3-1 \times 3 = -3).
3
Rewrite the fraction using the expanded denominator.
16a5b28a3b−3\frac{16a^5b^2}{8a^3b^{-3}}
This sets up the expression for application of the quotient rule.
4
Divide the numerical coefficients.
16÷8=216 \div 8 = 2
Constants are separated from variables and divided normally.
5
Apply the quotient rule to the variables with the base aa.
a5−3=a2a^{5-3} = a^2
When dividing like bases, subtract the exponent of the denominator from the exponent of the numerator.
6
Apply the quotient rule to the variables with the base bb.
b2−(−3)=b2+3=b5b^{2 - (-3)} = b^{2 + 3} = b^5
Subtracting a negative exponent is equivalent to adding a positive number.
7
Combine the simplified coefficient and variables.
2a2b52a^2b^5
Bringing the distinct evaluated parts back together provides the final simplified term.

Key Concept

To simplify a rational expression involving exponents, first use the power rule to eliminate grouping symbols, ensuring numerical coefficients are raised to the power. Then, divide the coefficients and use the quotient rule to subtract the denominator's exponents from the numerator's exponents for matching bases.
Estimated Time:1m 30s
Question 206Question →

Match each unsimplified algebraic expression on the left with its fully simplified equivalent form on the right. Assume pp and qq represent non-zero numbers.

Click a left item, then click its matching right item

Items

(4p5q−2)(2p−1q6) (4p^5 q^{-2})(2p^{-1} q^6)
15p7q23p2q−3 \frac{15p^7 q^2}{3p^2 q^{-3}}
(3p3q2)2 (3p^3 q^2)^2
(2p2q3)34p4q7 \frac{(2p^2 q^3)^3}{4p^4 q^7}

Matches

Show answer & explanation

Answer

The correct matches are: (4p^5 q^-2)(2p^-1 q^6) matches 8p^4 q^4; the fraction 15p^7 q^2 over 3p^2 q^-3 matches 5p^5 q^5; (3p^3 q^2)^2 matches 9p^6 q^4; and the fraction (2p^2 q^3)^3 over 4p^4 q^7 matches 2p^2 q^2.
Each expression simplifies to exactly one of the given target forms by consistently applying the laws of exponents (product, quotient, and power rules) and handling numerical coefficients accurately.

Step-by-Step Solution

1
Simplify the first expression using the product rule for exponents.
The coefficients multiply to 8. Adding the exponents for p gives 5 + (-1) = 4. Adding the exponents for q gives -2 + 6 = 4. The result is 8p^4 q^4.
When multiplying terms with the same base, you add their exponents.
2
Simplify the second expression using the quotient rule for exponents.
The coefficients divide to 5. Subtracting the p exponents gives 7 - 2 = 5. Subtracting the q exponents gives 2 - (-3) = 5. The result is 5p^5 q^5.
When dividing terms with the same base, you subtract the denominator's exponent from the numerator's exponent.
3
Simplify the third expression using the power of a product rule.
The coefficient squared is 3^2 = 9. Multiplying the p exponent by 2 gives 3 * 2 = 6. Multiplying the q exponent by 2 gives 2 * 2 = 4. The result is 9p^6 q^4.
When raising a power to a power, multiply the exponents. Remember to also apply the power to the numerical coefficient.
4
Simplify the fourth expression by applying the power rule first, then the quotient rule.
The numerator becomes 8p^6 q^9. Dividing this by 4p^4 q^7 gives a coefficient of 2, p exponent of 6 - 4 = 2, and q exponent of 9 - 7 = 2. The result is 2p^2 q^2.
Order of operations requires handling the exponent on the entire numerator before performing division.

Key Concept

Applying the product, quotient, and power rules of exponents to simplify algebraic expressions.
Question 207Question →

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

Show answer & explanation

Answer: 4p2+7p−24p^2 + 7p - 2

Answer

The equivalent expression is 4p2+7p−24p^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 4p24p^2, 8p8p, −p-p, and −2-2. Combining the like terms (8p8p and −p-p) yields 7p7p, making the final simplified expression 4p2+7p−24p^2 + 7p - 2.

Step-by-Step Solution

1
Multiply the first terms of each binomial.
4p×p=4p24p \times p = 4p^2
This establishes the quadratic term of the final polynomial.
2
Multiply the outer terms.
4p×2=8p4p \times 2 = 8p
This finds the first part of the linear middle term.
3
Multiply the inner terms.
−1×p=−p-1 \times p = -p
This finds the second part of the linear middle term.
4
Multiply the last terms.
−1×2=−2-1 \times 2 = -2
This finds the constant term of the final polynomial.
5
Combine the like terms.
4p2+(8p−p)−2=4p2+7p−24p^2 + (8p - p) - 2 = 4p^2 + 7p - 2
Simplifying the expression converts it into standard polynomial form.

Key Concept

Multiplying Binomials
Question 208Question →

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

Fill in the blanks below

A mechanic orders a set of spark plugs that cost 8each,plusaflatshippingfeeof8 each, plus a flat shipping fee of 12. If the total charge is $60, the mechanic ordered spark plugs.
Show answer & explanation

Answer

6
By setting up the equation 8x+12=608x + 12 = 60, we can isolate xx. Subtracting the constant 1212 from both sides gives 8x=488x = 48. Dividing by 88 yields x=6x = 6. Therefore, 66 spark plugs were ordered.

Step-by-Step Solution

1
Set up a linear equation to represent the word problem.
8x+12=608x + 12 = 60
Let xx be the number of spark plugs. Multiplying the cost per plug (8)by8) by x andaddingtheflatshippingfee( and adding the flat shipping fee ( 12) equals the total charge ($60).
2
Subtract 12 from both sides of the equation.
8x=488x = 48
To begin isolating the variable xx, the constant term must be removed from the left side of the equation.
3
Divide both sides by 8.
x=6x = 6
To completely isolate xx, divide by its coefficient. This reveals the number of spark plugs ordered.

Key Concept

Solving one-variable linear equations from a real-world scenario.
Question 209Question →

A customer's monthly utility bill is calculated as twelve dollars less than three times the sum of their base usage fee, ff, and five. If the total bill amounts to 108108, which of the following equations correctly models this calculation?

Show answer & explanation

Answer: 3(f+5)−12=1083(f + 5) - 12 = 108

Answer

The equation that correctly represents the calculation is 3(f+5)−12=1083(f + 5) - 12 = 108.
The equation is built step-by-step: 'the sum of their base usage fee, ff, and five' is written as (f+5)(f + 5). 'Three times' this sum requires multiplying the grouped expression, resulting in 3(f+5)3(f + 5). Finally, 'twelve dollars less than' means 1212 must be subtracted from that product, yielding the final correct equation 3(f+5)−12=1083(f + 5) - 12 = 108.

Step-by-Step Solution

1
Translate the phrase 'the sum of their base usage fee, ff, and five'.
(f+5)(f + 5)
The phrase 'the sum of' indicates addition between the variable ff and the number 55. Parentheses are required because a subsequent operation applies to this entire sum.
2
Translate 'three times the sum'.
3(f+5)3(f + 5)
The multiplier 33 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)−123(f + 5) - 12
The phrase 'less than' is a turnaround phrase meaning 1212 is subtracted from the quantity that precedes it conceptually, placing −12- 12 at the end.
4
Form the complete equation by setting the expression equal to the total bill.
3(f+5)−12=1083(f + 5) - 12 = 108
The phrase 'amounts to' signifies equality with the final given value of 108108.

Key Concept

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'.
Estimated Time:1m 0s
Question 210Question →

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

Click a left item, then click its matching right item

Items

The total cost of kk items priced at twelve dollars each, minus a five-dollar discount.
Twelve less than the quotient of a number kk and five.
The product of twelve and the quantity of five less than a number kk.
The area of a rectangle with length twelve and width kk, increased by five.

Matches

Show answer & explanation

Answer

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.

Step-by-Step Solution

1
Translate the rate and discount scenario.
The phrase 'kk items at twelve dollars each' translates to 12k12k. The phrase 'minus a five-dollar discount' makes it 12k−512k - 5.
Multiplication represents a per-item rate, and a discount implies subtraction.
2
Translate the quotient phrase.
The 'quotient of kk and five' is k5\frac{k}{5}. 'Twelve less than' means subtracting 1212 from this term, yielding k5−12\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 kk' is translated as (k−5)(k - 5). The 'product of twelve and the quantity' is 12(k−5)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 kk' translates to 12k12k. 'Increased by five' adds 55, yielding 12k+512k + 5.
Area of a rectangle is calculated by multiplying length and width, and 'increased by' signifies addition.

Key Concept

Translating Word Phrases into Algebraic Equations
Question 211Question →

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.

Click a left item, then click its matching right item

Items

(3a4b2)(4a2b)(3a^4b^2)(4a^2b)
24a7b36a3b6\frac{24a^7b^3}{6a^3b^6}
(2a3b−2)4(2a^3b^{-2})^4
(5a0b3)−2(5a^0b^3)^{-2}

Matches

Show answer & explanation

Answer

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.

Step-by-Step Solution

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.

Key Concept

Applying the product, quotient, power, zero, and negative rules of exponents to simplify algebraic expressions with multiple variables and coefficients.
Question 212Question →
What value of mm makes the equation 12m−7=41 12m - 7 = 41 true?
Show answer & explanation

Answer: 4

Answer

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

Step-by-Step Solution

1
Add 7 to both sides of the equation.
12m=48 12m = 48
This is the first inverse operation needed to isolate the variable term.
2
Divide both sides by 12.
m=4 m = 4
This inverse operation isolates mm, giving the final solution.

Key Concept

Solving two-step linear equations using inverse operations
Estimated Time:30s
Question 213Question →

Consider the algebraic expression below, where xx and yy represent non-zero values:

(4x3y−2)22x−1y3 \frac{(4x^3 y^{-2})^2}{2x^{-1} y^3}

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

Show answer & explanation

Answer: 8x7y7 \frac{8x^7}{y^7}

Answer

The completely simplified form is 8x7y7 \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 16x6y−416x^6y^{-4}. Next, dividing the coefficients yields 16÷2=816 \div 2 = 8. Then, the quotient rule is applied by subtracting the denominator's exponents from the numerator's exponents for matching bases: for xx, 6−(−1)=76 - (-1) = 7, and for yy, −4−3=−7-4 - 3 = -7. This results in 8x7y−78x^7y^{-7}. Finally, the negative exponent is resolved by moving y7y^7 to the denominator, giving the completely simplified form 8x7y7 \frac{8x^7}{y^7} .

Step-by-Step Solution

1
Apply the power of a product rule to simplify the numerator.
(4x3y−2)2=42⋅(x3)2⋅(y−2)2=16x6y−4 (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.
16x6y−42x−1y3 \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÷2=8 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 xx: 6−(−1)=6+1=7 6 - (-1) = 6 + 1 = 7 . For yy: −4−3=−7 -4 - 3 = -7 .
The quotient rule states that aman=am−n \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.
8x7y−7=8x7y7 8x^7y^{-7} = \frac{8x^7}{y^7}
A negative exponent represents the reciprocal of the base (a−n=1an a^{-n} = \frac{1}{a^n} ). Moving y−7y^{-7} to the denominator as y7y^7 yields the final simplified expression format.

Key Concept

Operations with Variable Exponents
Question 214Question →

Determine the missing coefficient in the area expression.

Fill in the blanks below

A rectangle has a length of (3x−4)(3x - 4) units and a width of (x+6)(x + 6) units. The area of the rectangle can be represented by the expanded polynomial $3x^2 + x - 24$ square units.
Show answer & explanation

Answer

14
The area of the rectangle is calculated by multiplying its length and width: (3x−4)(x+6)(3x - 4)(x + 6). Distributing the terms, the outer product is 18x18x and the inner product is −4x-4x. Combining these like terms (18x−4x18x - 4x) results in 14x14x. Therefore, the missing coefficient for the xx term is 14.

Step-by-Step Solution

1
Set up the multiplication for the area of the rectangle.
(3x−4)(x+6)(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.
3x2+18x−4x−243x^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 xx variable.
18x−4x=14x18x - 4x = 14x
Simplifying the expression requires combining terms that share the same variable and exponent.

Key Concept

Multiplying binomials and combining like terms
Question 215Question →

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

Show answer & explanation

Answer: m2−4m−12m^2 - 4m - 12

Answer

The correct answer is the polynomial m2−4m−12m^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 m2+2m−6m−12m^2 + 2m - 6m - 12. Combining the like terms (2m−6m2m - 6m) yields −4m-4m, resulting in the final simplified polynomial m2−4m−12m^2 - 4m - 12.

Step-by-Step Solution

1
Apply the distributive property to multiply each term in the first binomial by each term in the second binomial.
m⋅m+m⋅2−6⋅m−6⋅2m \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.
m2+2m−6m−12m^2 + 2m - 6m - 12
To simplify the expression before combining any like terms.
3
Combine the like terms (the middle terms containing mm).
m2−4m−12m^2 - 4m - 12
To write the polynomial in its standard, fully simplified form.

Key Concept

Multiplying Polynomials

Alternative Method

You can use the area model (or box method) to organize the multiplication. Draw a 2x2 grid, place the terms mm and −6-6 along the top edge, and mm and 22 along the side. Multiply the intersecting terms to fill the four cells (m2m^2, −6m-6m, 2m2m, and −12-12), and then sum all the cells to find the final polynomial.
Estimated Time:45s
Question 216Question →

The expression (2y−7)2(2y - 7)^2 is expanded into the standard polynomial form ay2+by+cay^2 + by + c. What is the value of the coefficient bb?

Show answer & explanation

Answer: -28

Answer

-28
Expanding the perfect square binomial (2y−7)2(2y - 7)^2 requires multiplying (2y−7)(2y - 7) by itself. Using the distributive property, the outer product is (2y)(−7)=−14y(2y)(-7) = -14y and the inner product is (−7)(2y)=−14y(-7)(2y) = -14y. Combining these like terms gives a total middle term of −28y-28y. Because the required polynomial form is ay2+by+cay^2 + by + c, the value of the coefficient bb is exactly -28.

Step-by-Step Solution

1
Rewrite the expression as a product of two binomials.
(2y−7)(2y−7)(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.
4y2−14y−14y+494y^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.
4y2−28y+494y^2 - 28y + 49
The middle terms, −14y-14y and −14y-14y, have the same variable raised to the same power, so their coefficients can be added together.
4
Identify the target coefficient.
−28-28
The standard polynomial form is given as ay2+by+cay^2 + by + c. The value of bb corresponds to the coefficient of the yy term.

Key Concept

Expanding Perfect Square Binomials
Question 217Question →

Given the expression 7x−2y27x - 2y^2, what is its numerical value when x=5x = 5 and y=3y = 3?

Show answer & explanation

Answer: 17

Answer

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

Step-by-Step Solution

1
Substitute the values x=5x = 5 and y=3y = 3 into the algebraic expression.
7(5)−2(3)27(5) - 2(3)^2
Variables must be replaced with their numerical values before evaluating.
2
Calculate the exponent.
7(5)−2(9)7(5) - 2(9)
According to the order of operations (PEMDAS), exponents must be evaluated before multiplication.
3
Perform the multiplication operations.
35−1835 - 18
Multiplication is evaluated before subtraction.
4
Subtract the resulting numbers.
17
This final arithmetic step yields the evaluated expression.

Key Concept

Evaluating Algebraic Expressions
Question 218Question →

Consider the linear equation below:

6y−18=24 6y - 18 = 24

What is the value of yy?

Show answer & explanation

Answer: 7

Answer

The correct value is 7.
The correct answer is found by first adding 18 to both sides of the equation to get 6y=426y = 42. Then, dividing both sides by 6 isolates yy, resulting in y=7y = 7.

Step-by-Step Solution

1
Isolate the term with the variable by adding 18 to both sides of the equation.
6y=42 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 y = 7
Dividing by the coefficient of the variable isolates yy and provides the final solution.

Key Concept

Solving linear equations by isolating the variable using inverse operations.
Question 219Question →

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

Show answer & explanation

Answer: C=500−2(w+50)C = 500 - 2(w + 50)

Answer

The correct equation is C=500−2(w+50)C = 500 - 2(w + 50).
The correct equation translates the text sequentially while respecting mathematical grouping. First, 'the sum of its base weight, ww, and fifty' translates to the grouped expression (w+50)(w + 50). 'Twice the sum' means multiplying that entire group by 2, yielding 2(w+50)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)500 - 2(w + 50).

Step-by-Step Solution

1
Translate the phrase 'the sum of its base weight, w, and fifty'.
(w+50)(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)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)C = 500 - 2(w + 50)
The phrase 'subtracted from X' means X is the starting amount. Therefore, the expression 2(w+50)2(w + 50) must be subtracted from 500.

Key Concept

Translating Word Phrases into Algebraic Equations
Question 220Question →

Find the value of tt that satisfies the linear equation below.

Fill in the blanks below

t3+8=15 \frac{t}{3} + 8 = 15

The value of tt is .
Show answer & explanation

Answer

21
To find the value of tt, you must apply inverse operations to isolate the variable. First, subtract 8 from both sides to get t/3=7t/3 = 7. Then, multiply both sides by 3 to eliminate the fraction, which gives the final solution t=21t = 21.

Step-by-Step Solution

1
Isolate the fraction term by subtracting 8 from both sides of the equation.
t3=15−8 \frac{t}{3} = 15 - 8
t3=7 \frac{t}{3} = 7
To solve for a variable, you must reverse the order of operations. Addition is reversed first using subtraction to isolate the term containing the variable.
2
Solve for tt by multiplying both sides of the equation by 3.
t=7×3 t = 7 \times 3
t=21 t = 21
Multiplying by the denominator cancels out the fraction, leaving the variable tt by itself.

Key Concept

Solving two-step linear equations using inverse operations.
PreviousPage 11 / 15Next
Algebra Practice Questions — Armed Services Vocational Aptitude Battery (ASVAB) — Page 11 | Examkin