Question

How do you factor \[10{x^2} - 7x - 12\]?

Answer 1

Hint: This problem deals with solving a quadratic equation. Here, given a quadratic equation expression, we have to simplify the expression and make it into a standard form of quadratic equation. If the quadratic equation is in the form of $a{x^2} + bx + c = 0$, then we know that the roots of this quadratic equation are given by :
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Complete step-by-step answer:
The given expression is a quadratic expression but not a quadratic equation. We have to convert the expression into an equation and then solve the quadratic equation.
The given expression is \[10{x^2} - 7x - 12\], consider it as given below:
$ \Rightarrow 10{x^2} - 7x - 12$
Now this is in the standard form of a quadratic expression, now comparing the coefficients $a,b$ and $c$:
$ \Rightarrow a = 10,b = - 7$ and $c = - 12$
Now applying the formula to find the value of the roots of $x$, as given below:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Substituting the values of $a,b$ and $c$ in the above formula:
$ \Rightarrow x = \dfrac{{ - \left( { - 7} \right) \pm \sqrt {{{\left( { - 7} \right)}^2} - 4\left( {10} \right)\left( { - 12} \right)} }}{{2\left( {10} \right)}}$
$ \Rightarrow x = \dfrac{{7 \pm \sqrt {49 + 480} }}{{20}}$
$ \Rightarrow x = \dfrac{{7 \pm \sqrt {529} }}{{20}}$
We know that the square root of 529 is 23, $\sqrt {529} = 23$
$ \Rightarrow x = \dfrac{{7 \pm 23}}{{20}}$
Now considering the two cases, with plus and minus, as shown:
$ \Rightarrow x = \dfrac{{7 + 23}}{{20}};x = \dfrac{{7 - 23}}{{20}}$
$ \Rightarrow x = \dfrac{{30}}{{20}};x = \dfrac{{ - 16}}{{20}}$
Hence the value of the roots are equal to :
$\therefore x = \dfrac{3}{2};x = \dfrac{{ - 4}}{5}$
Now factoring the given expression as \[10{x^2} - 7x - 12 = \left( {x - \dfrac{3}{2}} \right)\left( {x + \dfrac{4}{5}} \right)\]
$\therefore 10{x^2} - 7x - 12 = \left( {2x - 3} \right)\left( {5x + 4} \right)$

Final Answer: The factors of the expression \[10{x^2} - 7x - 12\] are $\left( {2x - 3} \right)$and $\left( {5x + 4} \right)$.

Note:
Please note that this problem can also be done either by the method of completing the square or just factoring and solving the quadratic equation. To solve $a{x^2} + bx + c = 0$ by completing the square: transform the equation so that the constant term, $c$ is alone on the right side. But here we are adding and subtracting some terms in order to factor.