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# How do you factor ${x^2} + 4x - 12$?

Last updated date: 01st Mar 2024
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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 – root1) (x – root 2).

Complete step-by-step solution:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factoring or by using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. So the equation is written as ${x^2} + 4x - 12$.
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=1 b=4 and c=-12. Now substituting these values to the formula for obtaining the roots we have
$roots = \dfrac{{ - 4 \pm \sqrt {{{(4)}^2} - 4(1)( - 12)} }}{{2(1)}}$
On simplifying the terms, we have
$\Rightarrow roots = \dfrac{{ - 4 \pm \sqrt {16 + 48} }}{2}$
Now add 16 to 48 we get
$\Rightarrow roots = \dfrac{{ - 4 \pm \sqrt {64} }}{2}$
The number 64 is a perfect square so we can take out from square root we have
$\Rightarrow roots = \dfrac{{ - 4 \pm 8}}{2}$
Therefore, we have $root1 = \dfrac{{ - 4 + 8}}{2} = \dfrac{4}{2} = 2$ or $root2 = \dfrac{{ - 4 - 8}}{2} = \dfrac{{ - 12}}{2} = - 6$.
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
$\Rightarrow \left( {x - 2} \right)\left( {x - ( - 6)} \right)$
On simplifying we have
$\Rightarrow (x - 2)(x + 6)$

Hence, we have found the factors for the given equation

Note: To find the factors for algebraic equations or expressions, it depends on the degree of the equation. If the equation contains a square then we have two factors. If the equation contains a cube then we have three factors. Here this equation also contains 3 factors, the two factors are imaginary.