If the expression (x + 4)^2 - (x - 4)^2 is equivalent to kx for all values of x, where k i… — SAT

If the expression $(x + 4)^2 - (x - 4)^2$ is equivalent to $kx$ for all values of $x$ , where $k$ is a constant, what is the value of $k$ ?

Answer: 16

The value of the constant

k

16

To find the value of

k

, the expression

(x + 4)^2 - (x - 4)^2

is simplified by expanding both binomials. The first binomial expands to

x^2 + 8x + 16

, and the second binomial expands to

x^2 - 8x + 16

. Subtracting the second expression from the first requires distributing the negative sign across all terms:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x - (-8x) + 16 - 16 = 16x

. Equating

16x

kx

yields

k = 16

Expand the first squared term,

(x + 4)^2

x^2 + 8x + 16

Applying the binomial expansion formula

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

where

a = x

and

b = 4

Expand the second squared term,

(x - 4)^2

x^2 - 8x + 16

Applying the binomial expansion formula

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

where

a = x

and

b = 4

Subtract the second expanded expression from the first and simplify

16x

Group like terms and distribute the negative sign:

(x^2 + 8x + 16) - (x^2 - 8x + 16) = x^2 - x^2 + 8x + 8x + 16 - 16 = 16x

Equate the simplified expression to

kx

to determine the constant

k

k = 16

Since

16x = kx

for all values of

x

, the coefficients must be equal.

Equivalent Algebraic Expressions

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