Question

How do you factor$2{x^2} - x - 15?$

Answer 1

The given question describes the operation of addition/ subtraction/ multiplication/ division. Also, remind the basic form of a quadratic equation and quadratic formula which is used to find the value of$x$. And, compare the given equation with the quadratic formula to solve the question. We need to know the root value of basic numbers.

Complete step by step solution:
The given equation is shown below,
$2{x^2} - x - 15 = 0 \to \left( 1 \right)$

We know that the quadratic formula is,
$a{x^2} + bx + c = 0 \to \left( 2 \right)$

Then, $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \to \left( 3 \right)$

Let’s compare the equation$\left( 1 \right)$and$\left( 2 \right)$, for finding the values
of$a,b$and$c$.

$\left( 1 \right) \to 2{x^2} - x - 15 = 0$
$\left( 2 \right) \to a{x^2} + bx + c = 0$

So, let’s compare the${x^2}$terms in the equation$\left( 1 \right)$and$\left( 2 \right)$

$$2 \times {x^2} \\\ a \times {x^2} \\\$$

So, we find$a = 2$.

Let’s compare the$x$terms in the equation$\left( 1 \right)$and$\left( 2 \right)$

$$\- 1 \times x \\\ b \times x \\\$$

So, we find$b = - 1$.

Let’s compare the constant terms in the equation$\left( 1 \right)$and$\left( 2 \right)$

So, we find$c = - 15$.

So, we get$a,b$and$c$values are$2, - 1$and$- 15$respectively.

Let’s substitute this value in the equation$\left( 3 \right)$for finding the value of$x$

$\left( 3 \right) \to x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

\times 2}}$$ By solving the above equation we get,

x = \dfrac{{1 \pm \sqrt {1 + 120} }}{4} \\
x = \dfrac{{1 \pm \sqrt {121} }}{4} \\

We know that$$\sqrt {121} = 11$$, so the above equation becomes, $$x = \dfrac{{1 \pm 11}}{4}$$ Due to the presence of$$ \pm $$we get two values for$$x$$. Case: 1 $$x = \dfrac{{1 + 11}}{4}$$ $$x = \dfrac{{12}}{4}$$ $$x = 3$$ Case: 2

x = \dfrac{{1 - 11}}{4} \\
x = \dfrac{{ - 10}}{4} \\
x = \dfrac{{ - 5}}{2} \\

In case: 1 we assume$$ \pm $$as an$$ + $$operation, in case: 2 we assume$$ \pm $$as an$$ - $$ operation. By considering the$$ \pm $$as$$ + $$we get $$x = 3$$ and by considering the$$ \pm $$as$$ - $$we get$$x = \dfrac{{ - 5}}{2}$$. **So, the final answer is, $$x = 3$$or$$x = \dfrac{{ - 5}}{2}$$.** **Note:** To find the value of$$x$$ from the given equation we would compare the equation with the quadratic formula. After comparing the equations we would find the value of$$a,b$$and$$c$$. When substituting these values in quadratic formula remind the following things, 1) When a negative number is multiplied with a negative number, the answer becomes a positive number. 2) When a positive number is multiplied with a positive number, the answer becomes a positive number. 3) When a negative number is multiplied with a positive number, the answer becomes negative.