Question

How do you solve ${{\left( x+1 \right)}^{2}}=25$?

Answer 1

Hint: In order to find the solution of this question, we will first subtract 25 from both sides of the equation then we will use the property ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$, then we will perform all the necessary calculations and simplify our answer to get the value of x.

Complete answer:
According to the question, we have been asked to find the value of x in equation ${{\left( x+1 \right)}^{2}}=25$.
To solve this question, we will start by subtracting 25 from both sides of the equation. Therefore, we get
${{\left( x+1 \right)}^{2}}-25=25-25$
Now, we know that the same terms with opposite signs cancel out. Therefore, we get
${{\left( x+1 \right)}^{2}}-25=0$
As we know that 25 is the perfect square of 5, that is, $5\times 5=25$. Hence, we can write the above equation as
${{\left( x+1 \right)}^{2}}-{{\left( 5 \right)}^{2}}=0$
Now, we will use the property, ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$. Therefore, for $a=\left( x+1 \right)$ and $b=5$, we get
${{\left( x+1 \right)}^{2}}-{{\left( 5 \right)}^{2}}=\left[ \left( x+1 \right)+5 \right]\left[ \left( x+1 \right)-5 \right]$
Now, we will simplify the above equation further. Therefore, we get
$\left( \left( x+1 \right)+5 \right)\left( \left( x+1 \right)-5 \right)=0$
And hence we get
$\left( x+6 \right)\left( x-4 \right)=0$
And we know that it can be further written as
$\left( x+6 \right)=0$ and $\left( x-4 \right)=0$
Which is the same as x = -6 and x = 4.
Therefore, we get the required value of x for ${{\left( x+1 \right)}^{2}}=25$ as -6 and 4.

Note: The other method to solve this question was by expanding the term ${{\left( x+1 \right)}^{2}}$ using the property $\left[ {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \right]$ and then taking 25 to the left-hand side and rearrange the terms to get the value of x using discriminant formula, that is \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. Also, we should be very careful while solving this question because if we make any type of calculation then we will end up with the wrong answer.