Question

How do you solve the equation ${{\left( 11x-5 \right)}^{2}}-{{\left( 10x-1 \right)}^{2}}-\left( 3x-20 \right)\left( 7x+10 \right)=124$

Answer 1

Hint: Now first we will expand the brackets in the equation with power 2. We know that ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ hence using this we will get a simplified equation. Now we will further use distributive property which says $a\left( b\pm c \right)=ab\pm ac$ . Now we will simplify the equation and get a linear equation. Now we will separate the variables and constants in the equation and solve to find the value of x.

Complete step by step solution:
Now consider the given equation ${{\left( 11x-5 \right)}^{2}}-{{\left( 10x-1 \right)}^{2}}-\left( 3x-20 \right)\left( 7x+10 \right)=124$
Here the given equation has terms with power 2. Hence first we will expand the squares of the terms and simplify to write the equation in general form.
Now we know that ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ Hence using this we get,
$\Rightarrow \left[ {{\left( 11x \right)}^{2}}-2\left( 11x \right)\left( 5 \right)+{{5}^{2}} \right]-\left[ {{\left( 10x \right)}^{2}}-2\left( 10x \right)\left( 1 \right)+{{\left( 1 \right)}^{2}} \right]-\left[ \left( 3x-20 \right)\left( 7x+10 \right) \right]=124$
Now on simplifying the above equation we get,
$\Rightarrow \left[ 121{{x}^{2}}-110x+25 \right]-\left[ 100{{x}^{2}}-20x+1 \right]-\left[ \left( 3x-20 \right)\left( 7x+10 \right) \right]=124$
Now on adding and subtracting the terms with same power we get, \[\Rightarrow 21{{x}^{2}}-90x+24-\left[ \left( 3x-20 \right)\left( 7x+10 \right) \right]=124\]
Now we know according to distributive property we have $a\left( b\pm c \right)=ab\pm ac$
Now using the distributive property we get,
$\begin{align}
  & \Rightarrow 21{{x}^{2}}-90x+24-\left[ \left( 3x-20 \right)\left( 7x \right)+\left( 3x-20 \right)\left( 10 \right) \right]=124 \\
 & \Rightarrow 21{{x}^{2}}-90x+24-\left[ 7x\left( 3x \right)-20\left( 7x \right)+10\left( 3x \right)-10\left( 20 \right) \right]=124 \\
 & \Rightarrow 21{{x}^{2}}-90x+24-\left[ 21{{x}^{2}}-140x+30x-200 \right]=124 \\
\end{align}$
Now again opening the bracket we get,
$\Rightarrow 21{{x}^{2}}-90x+24-21{{x}^{2}}+140x-30x+200=124$
Again adding and subtracting the terms with same power and simplifying we get the ab9ve equation as,
 $\Rightarrow 20x+224=124$
Now we will separate the constant and variable in the equation. Hence we get,
$\begin{align}
  & \Rightarrow 20x=124-224 \\
 & \Rightarrow 20x=-100 \\
\end{align}$
Now dividing the whole equation by 20 we get, x = - 5.
Hence the solution of the given equation is x = - 5.

Note:
Now always try to write the equation in general form first. Here we had a square in the equation but after simplification we get the equation as a linear equation. Also remember to always check the solution by substituting the obtained value in the equation.