Question

How do you factor completely $2{{x}^{2}}-18$?

Answer 1

Hint: In this problem we need to calculate the factors of the given equation. We can observe that the given equation is a quadratic equation in terms of $x$with the middle term or the $x$ term is zero. So, we will first make the coefficient of ${{x}^{2}}$ as $1$ by taking $2$. Now we will get an equation like ${{a}^{2}}-{{b}^{2}}$. Here we will apply the known algebraic formula which is ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$ and simplify the equation to get the required result.

Complete step by step solution:
Given equation, $2{{x}^{2}}-18$.
We can observe that the above equation is a quadratic equation which is in terms of $x$ and the middle term or the $x$ term is zero in the above equation.
Taking $2$ as common from the above equation, then we will get
$\Rightarrow 2{{x}^{2}}-18=2\left( {{x}^{2}}-9 \right)$
We can write the value $9$ as ${{3}^{2}}$, then the above equation is modified as
$\Rightarrow 2{{x}^{2}}-18=2\left( {{x}^{2}}-{{3}^{2}} \right)$
In the above equation we can observe the algebraic form ${{a}^{2}}-{{b}^{2}}$. In algebra we have the formula ${{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$. Applying this formula in the above equation, then we will get
$\Rightarrow 2{{x}^{2}}-18=2\left( x+3 \right)\left( x-3 \right)$
Hence the factors of the given equation $2{{x}^{2}}-18$ are $2$, $x+3$, $x-3$.

Note: In this problem we don’t have the middle term or $x$ term in the given quadratic equation so we have simplified factored by applying some arithmetic and algebraic formulas. If there is a middle term or $x$ term in the given quadratic equation, then we need to follow the factorization method which is splitting the middle term by considering the remaining coefficients in the equation.