Question

How do you factor \[2{x^2} + 5x - 6\]?

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 \[2{x^2} + 5x - 6\].
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=5 and c=-6. Now substituting these values to the formula for obtaining the roots we have
\[roots = \dfrac{{ - 5 \pm \sqrt {{5^2} - 4(2)( - 6)} }}{{2(2)}}\]
On simplifying the terms, we have
\[ \Rightarrow roots = \dfrac{{ - 5 \pm \sqrt {25 + 48} }}{4}\]
Now add 25 to 48 we get
\[ \Rightarrow roots = \dfrac{{ - 5 \pm \sqrt {73} }}{4}\]
The number 73 is not a perfect square so we can’t take out from square root we have
\[ \Rightarrow roots = \dfrac{{ - 5}}{4} \pm \dfrac{{\sqrt {73} }}{4}\]
Therefore, we have \[root_1 = \dfrac{{ - 5}}{4} + \dfrac{{\sqrt {73} }}{4}\] or \[root_2 = \dfrac{{ - 5}}{4} - \dfrac{{\sqrt {73} }}{4}\].
The roots for the quadratic equation when we find the roots by using formula is given by $(x – root_1)$ $(x – root_2)$
Substituting the roots values, we have
\[ \Rightarrow \left( {x - \left( { - \dfrac{5}{4} + \dfrac{{\sqrt {73} }}{4}} \right)} \right)\left( {x - \left( {\dfrac{{ - 5}}{4} - \dfrac{{\sqrt {73} }}{4}} \right)} \right)\]
On simplifying we have
\[ \Rightarrow \left( {x + \dfrac{5}{4} - \dfrac{{\sqrt {73} }}{4}} \right)\left( {x + \dfrac{5}{4} + \dfrac{{\sqrt {73} }}{4}} \right)\]
Hence, we have found the factors for the given equation
So, the correct answer is “$\left( {x + \dfrac{5}{4} - \dfrac{{\sqrt {73} }}{4}} \right)\left( {x + \dfrac{5}{4} + \dfrac{{\sqrt {73} }}{4}} \right)$”.

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.