Question

How do you factor completely \[2{x^2} - 8\] ?

Answer 1

Hint: Here in this question, we have to find the factors, the given equation is in the form of a quadratic equation. This is a quadratic equation for the variable x. By using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] , we can determine the roots of the equation and factors are given by $(x – root_1)$ $ (x – root _2)$

Complete step-by-step answer:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factorising or by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . So the equation is written as \[2{x^2} - 8\] .
In general, the quadratic equation is represented as \[a{x^2} + bx + c = 0\] , when we compare the above equation to the general form of equation the values are as follows. a=2 b=0 and c=-8. Now substituting these values to the formula for obtaining the roots we have
 \[roots = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4(2)( - 8)} }}{{2(2)}}\]
On simplifying the terms, we have
 \[ \Rightarrow roots = \dfrac{{0 \pm \sqrt {0 + 64} }}{4}\]
Now add 0 to 64 we get
 \[ \Rightarrow roots = \dfrac{{0 \pm \sqrt {64} }}{4}\]
The number 64 is a perfect square so we can take out from square root we have
 \[ \Rightarrow roots = \dfrac{{ \pm 8}}{4}\]
On simplifying we have
 \[ \Rightarrow roots = \pm 2\]
Therefore, we have \[root_1 = + 2\] or \[root_2 = - 2\] .
The roots for the quadratic equation when we find the roots by using formula is given by (x – root1) (x – root 2)
Substituting the roots values, we have
 \[ \Rightarrow \left( {x - 2} \right)\left( {x - ( - 2)} \right)\]
On simplifying we have
 \[ \Rightarrow \left( {x - 2} \right)\left( {x + 2} \right)\]
Hence, we have found the factors for the given equation
We can also solve this equation by using the standard algebraic formula \[{a^2} - {b^2} = (a + b)(a - b)\]
So, the correct answer is “$\left( {x - 2} \right)\left( {x + 2} \right)$”.

Note: The quadratic equation can be solved by using the factorisation method and we also find the roots by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] . While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.