Question

Solve the given algebraic expression: $4{x^2} - 9 = 0$

Answer 1

Hint:The given equation has the highest degree of two which means it is a quadratic equation. We can solve quadratic equations by two methods i.e. 1) Standard formula 2) Standard identity. We can use either of two to solve the given problem and find the values of the variable which is known as ‘roots of the equation’.

Complete solution step by step:
Firstly, we write down the equation $4{x^2} - 9 = 0$

Now we see that it is an equation with degree two (the highest power of the given polynomial equation is two). It also has the same form as the general form of quadratic equation i.e.
$a{x^2} + bx + c = 0;\;a,b\,{\text{and}}\,c \in \mathbb{R};\;a \ne 0$

Comparing the given equation with the standard equation gives us
$
a = 4 \\
b = 0 \\
c = 9 \\
$
Now we have to apply the standard formula to solve the equation and find the value of the variable.

The formula for finding the roots of a quadratic equation is given by
$x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$

Applying the quadratic formula, we get
\[
x = \dfrac{{0 \pm \sqrt {{{(0)}^2} - 4 \times 4 \times 9} }}{{2 \times (4)}} \\
\Rightarrow x = \dfrac{{\sqrt {144} }}{8} = \dfrac{3}{2} \\
\]

This is the root of the given equation which is the required solution.

We can also use the second method here to solve the problem i.e. using a standard identity of algebraic equation. The equation given in the question is of the form

${a^2} - {b^2} = (a + b) \times (a - b)$
So we convert our problem into this form
$
4{x^2} - 9 = 0 \\
\Rightarrow (2{x^2}) - {(3)^2} = 0 \\
\Rightarrow (2x + 3) \times (2x - 3) = 0 \\
$

Taking the two values equals to zero separately we get
$
(2x + 3) = 0 \\
\Rightarrow x = - \dfrac{3}{2} \\
{\text{and}} \\
(2x - 3) = 0 \\
\Rightarrow x = \dfrac{3}{2} \\
$

So we take the positive value in this case and thus it is our desired result.

Note: We can use both the methods in this question but if the problem is already in the standard form of algebraic identity, factoring method is appropriate to use. When a problem is not given in any standard form we can always use the quadratic formula to solve the question.