The expression (3x^2 + 5x) - (x^2 - kx), where k is a constant, can be rewritten as 2x^2 +… — SAT

The expression $(3x^2 + 5x) - (x^2 - kx)$ , where $k$ is a constant, can be rewritten as $2x^2 + 12x$ . What is the value of $k$ ?

Answer: 7

The value of

k

7

To find the value of

k

, we first simplify the expression

(3x^2 + 5x) - (x^2 - kx)

by distributing the subtraction sign to both terms inside the second set of parentheses. This yields

3x^2 + 5x - x^2 + kx

. Next, we group and combine like terms to get

(3x^2 - x^2) + (5x + kx) = 2x^2 + (5+k)x

. Since this expression is equivalent to

2x^2 + 12x

for all values of

x

, the coefficients of corresponding terms must be equal. Equating the coefficients of

x

gives

5+k = 12

. Subtracting 5 from both sides yields

k = 7

Distribute the negative sign to the terms in the second parentheses.

3x^2 + 5x - x^2 + kx

To remove the parentheses and simplify the expression.

Combine like terms.

2x^2 + (5 + k)x

Grouping the

x^2

terms and

x

terms simplifies comparison with the target expression.

Equate the coefficient of the

x

term to the corresponding coefficient in the target expression.

5 + k = 12 \implies k = 7

Equivalent expressions must have equal corresponding coefficients for all values of

x

Equivalence of polynomial expressions by combining like terms and equating coefficients

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