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# Solve the quadratic equation by factorization:-$64{x^2} - 9 = 0$

Last updated date: 20th Sep 2024
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Hint: We will use the identity related to (a – b) ( a + b) and use this by taking a = 8x and b = 3 and thus, we will have the factor of the given quadratic equation.

We are given that we need to solve $64{x^2} - 9 = 0$ using the method of factorization.
We can write the given quadratic equation as follows:-
$\Rightarrow {\left( {8x} \right)^2} - {3^2} = 0$
Now, we know that we have an identity: ${a^2} - {b^2} = (a - b)(a + b)$
T0 get the factors of the given equation, put a = 8x and b = 3 in the above given expression which is an identity to get the following expression:-
$\Rightarrow 64{x^2} - 9 = {\left( {8x} \right)^2} - {3^2} = \left( {8x + 3} \right)\left( {8x - 3} \right)$
Thus, we get the following equation:-
$\Rightarrow \left( {8x + 3} \right)\left( {8x - 3} \right) = 0$
Solving this, we get either $x = \dfrac{3}{8}$ or $x = - \dfrac{3}{8}$.

Hence, the answer is $x = \dfrac{3}{8}, - \dfrac{3}{8}$.

Note:
The students must note that solving an equation by using factorization helps us in many aspects because we do not have to use the formula for roots of a quadratic equation which itself involves a lot of calculations and can lead to mathematical errors.
The students must also note that in the last few steps, when we got the factors of the given quadratic equation, we did use a theorem to solve it.
It is as follows:- If a.b = 0, then either a = 0 or b = 0 or both a = b = 0.
We used this statement to get the required values of x, which is the roots of the given quadratic equation.
The students must note that an equation has as many roots as its degree. In quadratic equations, the degree is 2, so, we have 2 roots of a quadratic equation, they may be real and distinct as happened in the above question, they may be real and equal or they may be imaginary and exist in conjugates, this all depends upon the discriminant of the equation.