Question

Solve $25{x^2} - 36 = 0$?

Answer 1

Hint: For solving the quadratic equation, we can use the factorization method to obtain the solution of the variable. In the factorization method, we have to split the middle term. But in this question, the middle term is zero. Therefore, we can directly use the algebraic identity to solve the quadratic equation.

The algebraic identity used in this question is:
${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$

Complete step by step answer:
In this question, the given quadratic equation is $25{x^2} - 36 = 0$.
Let us apply the algebraic identity ${a^2} - {b^2} = \left( {a - b} \right)\left( {a + b} \right)$.
Now, compare the equation $25{x^2} - 36$with ${a^2} - {b^2}$.
We get the value of ‘a’ is equal to 5x and the value of ‘b’ is equal to 6.
Therefore,
$ \Rightarrow 25{x^2} - 36 = 0$
Let us apply the algebraic identity to the left-hand side.
$ \Rightarrow \left( {5x - 6} \right)\left( {5x + 6} \right) = 0$
That is equal to,
$ \Rightarrow \left( {5x - 6} \right) = 0$ And $ \Rightarrow \left( {5x + 6} \right) = 0$
First, we will solve,
$ \Rightarrow \left( {5x - 6} \right) = 0$
Let us add 6 on both sides.
$ \Rightarrow 5x - 6 + 6 = 0 + 6$
To solve the left-hand side, apply the addition. The addition of -6 and 6 is 0.
That is equal to,
$ \Rightarrow 5x = 6$
Divide both sides by 5.
$ \Rightarrow \dfrac{{5x}}{5} = \dfrac{6}{5}$
To solve the left-hand side, apply the division. The addition of 5 and 5 is 1.
Therefore,
$ \Rightarrow x = \dfrac{6}{5}$
Second, we will solve,
$ \Rightarrow \left( {5x + 6} \right) = 0$
Let us subtract 6 on both sides.
$ \Rightarrow 5x - 6 + 6 = 0 - 6$
To solve the left-hand side, apply the addition. The addition of -6 and 6 is 0.
That is equal to,
$ \Rightarrow 5x = - 6$
Divide both sides by 5.
$ \Rightarrow \dfrac{{5x}}{5} = - \dfrac{6}{5}$
To solve the left-hand side, apply the division. The addition of 5 and 5 is 1.
Therefore,
$ \Rightarrow x = - \dfrac{6}{5}$
Hence, the solution of the given equation is $x = \dfrac{6}{5}$ and $x = - \dfrac{6}{5}$.

Note: Alternate method to solve this question.
$ \Rightarrow 25{x^2} - 36 = 0$
Let us add 36 on both sides.
$ \Rightarrow 25{x^2} - 36 + 36 = 0 + 36$
To solve the left-hand side, apply the addition. The addition of -36 and 36 is 0.
Therefore,
$ \Rightarrow 25{x^2} = 36$
Now, divide both sides by 25.
$ \Rightarrow \dfrac{{25{x^2}}}{{25}} = \dfrac{{36}}{{25}}$
To solve the left-hand side, apply the division. The addition of 25 and 25 is 1.
So,
$ \Rightarrow {x^2} = \dfrac{{36}}{{25}}$
Now, let us apply the square root on both sides.
$ \Rightarrow \sqrt {{x^2}} = \sqrt {\dfrac{{36}}{{25}}} $
The square root of ${x^2}$ is x, the square root of 36 is 6, and the square root of 25 is 5.
Substitute all these values in the above equation. We will get,
 $ \Rightarrow x = \pm \dfrac{6}{5}$