If the expression (x + 5)^2 - (x - 3)^2 is equivalent to ax + b for all values of x, where… — SAT

If the expression $(x + 5)^2 - (x - 3)^2$ is equivalent to $ax + b$ for all values of $x$ , where $a$ and $b$ are constants, what is the value of $a$ ?

Answer: 16

Answer

Expanding

(x + 5)^2

gives

x^2 + 10x + 25

, and expanding

(x - 3)^2

gives

x^2 - 6x + 9

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

x^2 + 10x + 25 - (x^2 - 6x + 9) = x^2 + 10x + 25 - x^2 + 6x - 9

. Combining like terms yields

16x + 16

. Comparing this to

ax + b

shows that the coefficient of

x

a

, is 16.

Step-by-Step Solution

Expand the first squared binomial term

(x + 5)^2 = x^2 + 10x + 25

To express the binomial square as a trinomial using the perfect square identity

(u + v)^2 = u^2 + 2uv + v^2

Expand the second squared binomial term

(x - 3)^2 = x^2 - 6x + 9

To express the binomial square as a trinomial using the perfect square identity

(u - v)^2 = u^2 - 2uv + v^2

Subtract the expanded expressions and distribute the negative sign

x^2 + 10x + 25 - x^2 + 6x - 9

To combine the terms while correctly applying the distributive property to the subtracted expression.

Combine like terms to simplify the polynomial

16x + 16

To find the final simplified polynomial of the form

ax + b

Compare the simplified expression to the standard form to find the value of

a

a = 16

The constant

a

represents the coefficient of the linear term

x

, which is 16.

Key Concept

Simplifying algebraic expressions by expanding binomial products and combining like terms.

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