For all real numbers a and b, which of the following expressions are equivalent to (a + b)… — GRE General Test

For all real numbers $a$ and $b$ , which of the following expressions are equivalent to $(a + b)^2 - (a - b)^2$ ? Select all that apply.

The expressions

4ab

and

(2a)(2b)

are equivalent to

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

Expanding the squared expressions gives

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

and

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

. Subtracting the second expression from the first yields

(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = 4ab

. Since

(2a)(2b)

also equals

4ab

, both

4ab

and

(2a)(2b)

are mathematically equivalent to the original expression.

Expand

(a + b)^2

using the binomial square formula.

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

Squaring a sum produces a trinomial containing two squared terms and a middle product term.

Expand

(a - b)^2

using the binomial square formula.

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

Squaring a difference produces a trinomial with a negative middle product term.

Subtract the second expanded trinomial from the first.

(a^2 + 2ab + b^2) - (a^2 - 2ab + b^2) = a^2 - a^2 + 2ab + 2ab + b^2 - b^2 = 4ab

Distributing the negative sign across

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

changes the signs inside the second set of parentheses.

Compare the simplified result

4ab

against all given choices.

Both

4ab

and

(2a)(2b) = 4ab

are identical to the calculated result.

Algebraic multiplication shows

(2a)(2b) = 4ab

Binomial expansion and combination of like terms

Question