Question

How do you factor $256{x^2} - 49?$

Answer 1

Hint: This problem deals with factoring an algebraic expression. To factorize an algebraic expression, the highest common factors of the terms of the given algebraic expression are determined and then we group the terms accordingly. In simple terms, the reverse process of expansion of an algebraic expression is its factorization.

Complete step-by-step solution:
Given a quadratic expression and we are asked to factorize it.
The quadratic expression is given by $256{x^2} - 49$,
Now consider the quadratic expression and equate it to zero, as given below:
$ \Rightarrow 256{x^2} - 49 = 0$
Now solve the above quadratic equation to get the value of $x$, as shown below.
Rearranging the above quadratic equation, as given below:
$ \Rightarrow 256{x^2} = 49$
$ \Rightarrow {x^2} = \dfrac{{49}}{{256}}$
$ \Rightarrow x = \pm \sqrt {\dfrac{{49}}{{256}}} $
We know that the square roots of 49 and 256 are $\sqrt {49} = 7$ and $\sqrt {256} = 16$, hence substituting these values in the above equation, as shown below:
$ \Rightarrow x = \pm \dfrac{7}{{16}}$
$\therefore x = \dfrac{7}{{16}}$ and $x = \dfrac{{ - 7}}{{16}}$
Consider $x = \dfrac{7}{{16}}$, as shown below:
$ \Rightarrow x = \dfrac{7}{{16}}$
$ \Rightarrow \left( {x - \dfrac{7}{{16}}} \right) = 0$
$ \Rightarrow 16x - 7 = 0$
Now consider $x = \dfrac{{ - 7}}{{16}}$, as shown below:
$ \Rightarrow x = \dfrac{{ - 7}}{{16}}$
$ \Rightarrow \left( {x + \dfrac{7}{{16}}} \right) = 0$
$ \Rightarrow 16x + 7 = 0$
As $\left( {16x - 7} \right)\left( {16x + 7} \right) = 0$
Hence the expression \[256{x^2} - 49\] is equal to $\left( {16x - 7} \right)\left( {16x + 7} \right)$
$\therefore 256{x^2} - 49 = \left( {16x - 7} \right)\left( {16x + 7} \right)$

The factors of the expression $256{x^2} - 49$ is equal to $\left( {16x - 7} \right)\left( {16x + 7} \right)$.

Note: Please note that there are 2 other ways, that this problem can be solved. These two methods are described here. The first method is that after equating the given quadratic expression to 0, we can apply the formula of solving the roots of the quadratic equation to get the values of $x$, and then factorize the given expression from the obtained roots of the quadratic equation. Another method is that considering the given expression $256{x^2} - 49$, adding and subtracting this expression with the term $112x$, then the expression becomes $256{x^2} + 112x - 112x - 49$, then we can factorize the expression from here, we can obtain $256{x^2} + 112x - 112x - 49 = \left( {16x + 7} \right)\left( {16x - 7} \right)$.