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

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

Explanation

Solution

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 a2−b2=(a+b)(a−b){{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 (x+1)2=25{{\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
(x+1)2−25=25−25{{\left( x+1 \right)}^{2}}-25=25-25
Now, we know that the same terms with opposite signs cancel out. Therefore, we get
(x+1)2−25=0{{\left( x+1 \right)}^{2}}-25=0
As we know that 25 is the perfect square of 5, that is, 5×5=255\times 5=25. Hence, we can write the above equation as
(x+1)2−(5)2=0{{\left( x+1 \right)}^{2}}-{{\left( 5 \right)}^{2}}=0
Now, we will use the property, a2−b2=(a+b)(a−b){{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right). Therefore, for a=(x+1)a=\left( x+1 \right) and b=5b=5, we get
(x+1)2−(5)2=[(x+1)+5][(x+1)−5]{{\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
((x+1)+5)((x+1)−5)=0\left( \left( x+1 \right)+5 \right)\left( \left( x+1 \right)-5 \right)=0
And hence we get
(x+6)(x−4)=0\left( x+6 \right)\left( x-4 \right)=0
And we know that it can be further written as
(x+6)=0\left( x+6 \right)=0 and (x−4)=0\left( x-4 \right)=0
Which is the same as x = -6 and x = 4.
Therefore, we get the required value of x for (x+1)2=25{{\left( x+1 \right)}^{2}}=25 as -6 and 4.

Note: The other method to solve this question was by expanding the term (x+1)2{{\left( x+1 \right)}^{2}} using the property [(a+b)2=a2+b2+2ab]\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=−b±b2−4ac2ax=\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.