Question

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

Answer 1

Hint: If you carefully observe the expression given in the above problem, you may find that the expression ${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}$ is written in the form of ${{a}^{2}}-{{b}^{2}}$. And here, $a=\left( 2x+5 \right)$ and $b=\left( 2x-5 \right)$. We know the algebraic identity which says that: ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ using this identity in the given expression we can simplify it further to get the simplest form of the given expression.

Complete step by step answer:
The expression given in the above problem which we are asked to simplify is as follows:
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}$
The above expression is written in the following algebraic form:
${{a}^{2}}-{{b}^{2}}$
Here, $a=\left( 2x+5 \right)$ and $b=\left( 2x-5 \right)$. Now, to simplify this expression we can apply the following algebraic identity:
${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$
Now, substituting $a=\left( 2x+5 \right)$ and $b=\left( 2x-5 \right)$ in the above equation and we get,
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}=\left( 2x+5+2x-5 \right)\left( 2x+5-\left( 2x-5 \right) \right)$
In the R.H.S of the above equation, we are going to open the bracket written inside the second bracket and we get,
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}=\left( 2x+5+2x-5 \right)\left( 2x+5-2x+5 \right)$
Now, in the R.H.S of the above equation, in first bracket, 5 will get cancelled out and in the second bracket, $2x$ will get cancelled out and we are left with:
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}=\left( 2x+2x \right)\left( 5+5 \right)$
Adding the terms written inside the brackets which are on the R.H.S of the above equation we get,
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}=\left( 4x \right)\left( 10 \right)$
Now, multiplying 4 by 10 in the R.H.S of the above equation we get,
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}=40x$

From the above solution, we have simplified the given expression to $40x$.

Note: The alternate way of solving the above problem is to open the square of each of the two brackets as follows:
${{\left( 2x+5 \right)}^{2}}-{{\left( 2x-5 \right)}^{2}}$
The algebraic identity we are going to use in the above expression are as follows:
$\begin{align}
  & {{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}} \\
 & {{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}} \\
\end{align}$
Applying the first equation for ${{\left( 2x+5 \right)}^{2}}$ and second equation for ${{\left( 2x-5 \right)}^{2}}$ and we get,
$\begin{align}
  & {{\left( 2x+5 \right)}^{2}}={{\left( 2x \right)}^{2}}+2\left( 2x \right)\left( 5 \right)+{{5}^{2}} \\
 & {{\left( 2x-5 \right)}^{2}}={{\left( 2x \right)}^{2}}-2\left( 2x \right)\left( 5 \right)+{{5}^{2}} \\
\end{align}$
Now, substituting the above expressions in the given expression and we get,
$\begin{align}
  & \left( {{\left( 2x \right)}^{2}}+2\left( 2x \right)\left( 5 \right)+{{5}^{2}} \right)-\left( {{\left( 2x \right)}^{2}}-2\left( 2x \right)\left( 5 \right)+{{5}^{2}} \right) \\
 & =4{{x}^{2}}+20x+25-\left( 4{{x}^{2}}-20x+25 \right) \\
\end{align}$
Opening the bracket written after negative sign in the above expression we get,
$4{{x}^{2}}+20x+25-4{{x}^{2}}+20x-25$
In the above, $4{{x}^{2}}\And 25$ will get cancelled out and we are left with:
$40x$