Question

Simplify: ${\left(2x+5\right)}^2-{\left(2x-5\right)}^2$.

Answer 1

Simplification of algebraic expressions.

Step 1: Simplify the term ${\left(2x+5\right)}^2$.

Use the identity ${\left(a+b\right)}^2=a^2+b^2+2ab$ to simplify ${\left(2x+5\right)}^2$:

$\begin{array}{rcl}{\left(2x+5\right)}^2& =& {\left(2x\right)}^2+{\left(5\right)}^2+2\left(2x\right)\left(5\right)\\ & =& 4x^2+25+20x\end{array}$

Step 2: Simplify the term ${\left(2x-5\right)}^2$.

Use the identity ${\left(a-b\right)}^2=a^2+b^2-2ab$ to simplify ${\left(2x-5\right)}^2$:

$\begin{array}{rcl}{\left(2x-5\right)}^2& =& {\left(2x\right)}^2+{\left(5\right)}^2-2\left(2x\right)\left(5\right)\\ & =& 4x^2+25-20x\end{array}$

Step 3: Simplify the expression ${\left(2x+5\right)}^2-{\left(2x-5\right)}^2$.

Use the results obtained in Step 1 and 2 to simplify the given expression:

$\begin{array}{rcl}{\left(2x+5\right)}^2-{\left(2x-5\right)}^2& =& \left(4x^2+25+20x\right)-\left(4x^2+25-20x\right)\\ & =& 4x^2+25+20x-4x^2-25+20x\\ & =& \left(4x^2-4x^2\right)+\left(25-25\right)+\left(20x+20x\right)\left[\mathrm{Collect}\mathrm{like}\mathrm{terms}\mathrm{together}\right]\\ & =& 0+0+40x\\ & =& 40x\end{array}$

Hence, ${\left(2x+5\right)}^2-{\left(2x-5\right)}^{2}is40x$.