Question

How do you factor and solve $64{x^2} - 1 = 0?$

Answer 1

Hint: This problem deals with solving the given by factoring and then solving for $x$. This can 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.

Complete step-by-step solution:
Given the quadratic equation $64{x^2} - 1 = 0$, consider it as given below:
$ \Rightarrow 64{x^2} - 1 = 0$
Now adding the above quadratic equation with the term $8x$ and subtracting with the same term, as shown below:
\[ \Rightarrow 64{x^2} + 8x - 8x - 1 = 0\]
Here taking the term $8x$ common from the first two terms, and taking -1 common from the last two terms as shown below:
\[ \Rightarrow 8x\left( {8x + 1} \right) - 1\left( {8x + 1} \right) = 0\]
Now here taking the term \[\left( {8x + 1} \right)\] common from the above equation, as shown below:
\[ \Rightarrow \left( {8x - 1} \right)\left( {8x + 1} \right) = 0\]
Hence factorized the given quadratic equation, $64{x^2} - 1 = 0$, as given below:
$ \Rightarrow 64{x^2} - 1 = \left( {8x - 1} \right)\left( {8x + 1} \right)$
Hence the factors of the given quadratic equation, $64{x^2} - 1 = 0$ are $\left( {8x - 1} \right)$ and $\left( {8x + 1} \right)$.
Now solving for the solution of $x$, as given below:
Equating the obtained factors to zero, as shown below:
\[ \Rightarrow \left( {8x - 1} \right)\left( {8x + 1} \right) = 0\]
First consider the first factor and equate it to zero, as shown below:
\[ \Rightarrow \left( {8x - 1} \right) = 0\]
\[ \Rightarrow x = \dfrac{1}{8}\]
Now considering the second factor and equate it to zero, as shown below:
\[ \Rightarrow \left( {8x + 1} \right) = 0\]
\[ \Rightarrow x = \dfrac{{ - 1}}{8}\]
Hence the values of $x$ are $\dfrac{1}{8},\dfrac{{ - 1}}{8}$.

The factors of $64{x^2} - 1 = 0$ are \[\left( {8x - 1} \right)\left( {8x + 1} \right) = 0\], and the values of $x$ are $\dfrac{1}{8},\dfrac{{ - 1}}{8}$.

Note: Please note that this problem can also be solved by another method, which is described here. Instead of first factoring and then solving for $x$, we can directly the value of $x$ from the given equation $64{x^2} - 1 = 0$, this can be done by sending the constant 1 to the right hand side of the equation and then solve for $x$, and then factorize with the obtained solutions.