Question

How do you solve \[{{x}^{2}}-20x=0\] by factoring ?

Answer 1

Hint: A quadratic equation can be solved various ways like, by factorization, by Sridhar Acharya’s method, etc. Factorization method involves breaking down the quadratic expression into its linear factors and then, equating each linear factor to 0 to get various values of \[x\]. These values are nothing but the roots of the quadratic equation. So, we can start by taking \[x\] as common and then equating each of the products to 0.

Complete step by step answer:
The given equation is
\[{{x}^{2}}-20x=0\].... Equation 1
Over here, the quadratic expression is \[{{x}^{2}}-20x\]. If we take x common from both the terms, we get
\[x\left( x-20 \right)\]
Therefore, equation 1 becomes
\[x\left( x-20 \right)=0\]
Here, \[x\] and \[\left( x-20 \right)\] are the two linear factors of the quadratic expression \[x\left( x-20 \right)\]. These two factors cannot be simultaneously 0. Therefore, we have to equate the terms to 0 one at a time, keeping the other factor non-zero at that time.
We start off with taking \[x=0\] . This means that \[x=0\] is a root of equation 1.
Secondly, we take the second factor,
\[\begin{align}
  & x-20=0 \\
 & \Rightarrow x=20 \\
\end{align}\]
This means,\[20\] is also a factor of equation 1.

Therefore, we can conclude that \[x=0\] and \[x=20\] are the roots of the quadratic equation \[{{x}^{2}}-20x=0\] .

Note: If the constant term in the quadratic equation would have been some non zero term then, factors could have been found out by any one method out of the two. The first method is rewriting the coefficient of the linear term of the quadratic expression as sum of two numbers whose product is ac (if the quadratic expression is of the form \[a{{x}^{2}}+bx+c=0\]). The second method is the vanishing factor method. Here, a number, say d is chosen out of trial and error which is a root of the quadratic equation. Then, \[x-d\]will be a factor of the equation. Dividing the equation by this factor yields other factors.