Question

How do you factor the expression $4{x^2} - 1$?

Answer 1

Hint: Firstly, we will rewrite the given equation in general form i.e. $a{x^2} + bx + c = 0$ then, we will apply the middle term splitting method as it is easily applicable here i.e. we will split the middle term. After that we will find the factors of the given equation i.e. $4{x^2} - 1$.

Complete Step by Step Solution:
The given equation is $4{x^2} + 0x - 1 = 0$
Now, multiply the coefficient of ${x^2}$ with coefficient of ${x^0}$ , we get $4 \times \left( { - 1} \right) = - 4$
Positive factors of -4 are 1, 2 and 4
Now, we will subtract the positive factors of -4 in such a way that the difference will become the coefficient of $x$
Now, we will subtract 2 from 2
$ \Rightarrow 4{x^2} + 0x - 1 = 0$
$ \Rightarrow 4{x^2} + \left( {2 - 2} \right)x - 1 = 0$
$ \Rightarrow 4{x^2} + 2x - 2x - 1 = 0$
Now, we will take 2x common from the first two terms and -1 from the last two terms i.e.
$ \Rightarrow 2x\left( {2x + 1} \right) - 1\left( {2x + 1} \right) = 0$
$ \Rightarrow \left( {2x - 1} \right)\left( {2x + 1} \right) = 0$
Hence, factors of $4{x^2} + 0x - 1 = 0$ are $\left( {2x - 1} \right)$ and $\left( {2x + 1} \right)$
Now, either $\left( {2x - 1} \right) = 0$ or $\left( {2x + 1} \right) = 0$

So, we got two values of x i.e. $x = \dfrac{1}{2},\dfrac{{ - 1}}{2}$

Note:
There is an alternative method to solve the given equation in which we use Quadratic Formula i.e. $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ where $a$ is coefficient of ${x^2}$ , $b$ is coefficient of $x$ and $c$ is coefficient of ${x^0}$
Quadratic Formula is $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
The given equation is $4{x^2} + 0x - 1 = 0$
Here $a = 4$ , $b = 0$ and $c = - 1$
Hence, $x = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4\left( 4 \right)\left( { - 1} \right)} }}{{2\left( 4 \right)}}$
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {0 + 16} }}{8}$
On further simplification,
$ \Rightarrow x = \dfrac{{ - 0 \pm \sqrt {16} }}{8}$
Now, one value of $x = \dfrac{{ - 0 + 4}}{8}$ and another value of $x = \dfrac{{ - 0 - 4}}{8}$ i.e.
$ \Rightarrow x = \dfrac{4}{8}$ and $x = \dfrac{{ - 4}}{8}$
$ \Rightarrow x = \dfrac{1}{2},\dfrac{{ - 1}}{2}$
Hence, factors of given equation are $\left( {x - \dfrac{1}{2}} \right)$ and $\left( {x + \dfrac{1}{2}} \right)$.