Question

How do you solve \[{{\left( 5x+5 \right)}^{2}}=100\]?

Answer 1

Hint: This question is from the topic of algebra. In this question, we are going to find out the value of x. In solving this question, we will first square the term \[\left( 5x+5 \right)\] using the formula \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]. After that, we will solve the further question and get our answer. After that, we will see an alternate method to solve this question.

Complete step by step answer:
Let us solve this question.
In this question, we have asked to solve the equation \[{{\left( 5x+5 \right)}^{2}}=100\]. Or, we can say that we have to find the value of x from the given equation.
So, using the formula: \[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\], we can write the square of the term \[\left( 5x+5 \right)\] as
\[{{\left( 5x+5 \right)}^{2}}={{\left( 5x \right)}^{2}}+2\times 5x\times 5+{{\left( 5 \right)}^{2}}\]
The above equation can also be written as
\[\Rightarrow {{\left( 5x+5 \right)}^{2}}=25{{x}^{2}}+50x+25\]
Using the above equation, we can write the equation \[{{\left( 5x+5 \right)}^{2}}=100\] as
\[\Rightarrow 25{{x}^{2}}+50x+25=100\]
The above equation can also be written as
\[\Rightarrow 25{{x}^{2}}+50x+25-100=0\]
 \[\Rightarrow 25{{x}^{2}}+50x-75=0\]
The above equation can also be written as’
\[\Rightarrow 25\times {{x}^{2}}+25\times 2x-25\times 3=0\]
Taking 25 as common, we will get
\[\Rightarrow 25\left( {{x}^{2}}+2x-3 \right)=0\]
Now, dividing 25 to the both side of equation, we will get
\[\Rightarrow {{x}^{2}}+2x-3=0\]
The above equation can also be written as
\[\Rightarrow {{x}^{2}}+3x-x-3=0\]
Taking common as ‘x’ in the first two terms and taking common as ‘negative (-)’ in the other two terms, we will get
\[\Rightarrow x\left( x+3 \right)-\left( x+3 \right)=0\]
The above can also be written as
\[\Rightarrow \left( x+3 \right)\left( x-1 \right)=0\]
From the above, we can say that the \[x+3=0\] and \[x-1=0\]. Or, we can say that \[x=-3\] and \[x=1\].
So, now we have the equation \[{{\left( 5x+5 \right)}^{2}}=100\] and got the value of x as -3 and 1.

Note: We should have a better knowledge in the topic of algebra. We should remember the following formula:
\[{{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]
We can solve this question by an alternate method.
For that method, we should use a formula. The formula is: \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
So, the equation \[{{\left( 5x+5 \right)}^{2}}=100\] can also be written as
\[{{\left( 5x+5 \right)}^{2}}-100=0\]
The above can also be written as
\[\Rightarrow {{\left( 5x+5 \right)}^{2}}-{{\left( 10 \right)}^{2}}=0\]
Using the formula: \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\], we can write the above equation as
\[\Rightarrow \left( 5x+5+10 \right)\left( 5x+5-10 \right)=0\]
The above can also be written as
\[\Rightarrow \left( 5x+15 \right)\left( 5x-5 \right)=0\]
The above can also be written as
\[\Rightarrow \left[ 5\left( x+3 \right) \right]\left[ 5\left( x-1 \right) \right]=0\]
The above equation can also be written as
\[\Rightarrow \left( x+3 \right)\left( x-1 \right)=0\]
From the above, we will equate both the factors with zero and find the value of x as -3 and 1.
As we got the same answer from this method, so we can use this method too to solve this type of question.