Question

How do you factor completely: \[100{{x}^{2}}-1\]?

Answer 1

Hint: From the question given, we have been asked to factor completely the given expression \[100{{x}^{2}}-1\]. We can factor completely the given quadratic expression \[100{{x}^{2}}-1\] by transforming the given expression into a simplified form and then we have to check whether it is in the type of basic formula of algebra or not. We can solve the given question that is factor completely the given quadratic expression by using the basic formula of algebra. First of all, as we have been discussed earlier, we have to transform the given quadratic expression.

Complete step by step answer:
From the question given, we have been given that \[100{{x}^{2}}-1\]
We know that \[100{{x}^{2}}\] can be written as \[\Rightarrow {{\left( 10x \right)}^{2}}\].
By writing \[100{{x}^{2}}\] as \[\Rightarrow {{\left( 10x \right)}^{2}}\], we get the expression as, \[\Rightarrow {{\left( 10x \right)}^{2}}-1\]
We can also write it as \[\Rightarrow {{\left( 10x \right)}^{2}}-{{\left( 1 \right)}^{2}}\]
We can clearly observe that, the above expression is in the form of \[{{a}^{2}}-{{b}^{2}}\].
In algebra, we have one basic formula regarding the above expression.
The basic formula in algebra regarding the above expression is, \[{{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)\]
By comparing the formula and the expression, we get
\[\begin{align}
  & a=10x \\
 & b=1 \\
\end{align}\]
Substitute the above values in the formula we have written above.
By substituting the values in the formula which is written above, we get \[\Rightarrow {{\left( 10x \right)}^{2}}-{{\left( 1 \right)}^{2}}=\left( 10x+1 \right)\left( 10x-1 \right)\]
Therefore, \[\Rightarrow 100{{x}^{2}}-1=\left( 10x+1 \right)\left( 10x-1 \right)\]
Hence, the given quadratic expression is completely factored.

Note: We should be very careful while converting the given equation into more simplified form. Also, we should be well known about the basic formulae of algebra. We should be able to convert the given equation into the suitable formulae. Also, we should be very careful while applying the formula. This question can also be answered using the formulae for finding the roots of the quadratic equation in the form of $a{{x}^{2}}+bx+c=0$ is given by $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . For this question it will be $100{{x}^{2}}-1$ and the roots are $\dfrac{-0\pm \sqrt{{{0}^{2}}-4\left( 100 \right)\left( -1 \right)}}{2\left( 100 \right)}=\dfrac{\pm \sqrt{400}}{200}\Rightarrow \dfrac{\pm 20}{200}=\pm \dfrac{1}{10}$. Therefore the roots are $\dfrac{-1}{10},\dfrac{1}{10}$ and the factors are $\left( 10x+1 \right)$ and $\left( 10x-1 \right)$ .