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# How do you write $9{x^2} - 4$ in factored form?

Last updated date: 22nd Feb 2024
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Hint: We can solve this using algebraic identities. We use the identity ${a^2} - {b^2} = (a - b)(a + b)$ to solve the given problem. We can see that 49 and 36 are perfect squares. We can convert the given problem into ${a^2} - {b^2}$, since the square of 3 is 9 and square of 2 is 4.

Complete step-by-step solution:
Given, $9{x^2} - 4$
We can rewrite it as $= {3^2}.{x^2} - {2^2}$
$= {(3x)^2} - {2^2}$.
That is it is in the form ${a^2} - {b^2}$, where $a = 3x$ and $b = 2$.
We have the formula ${a^2} - {b^2} = (a - b)(a + b)$.
Then above becomes,
$= (3x - 2)(3x + 2)$. These are the factors of the $9{x^2} - 4$.

$\Rightarrow (3x - 2)(3x + 2) = 0$
$\Rightarrow 3x - 2 = 0$ and $3x + 2 = 0$.
$\Rightarrow 3x = 2$ and $3x = - 2$.
$\Rightarrow x = \dfrac{2}{3}$ and $x = - \dfrac{2}{3}$. This is the roots of the given polynomial.
Note: Follow the same procedure for these kinds of problems. Since the given equation is a polynomial. The highest exponent of the polynomial in a polynomial equation is called its degree. A polynomial equation has exactly as many roots as its degree. Here the degree is 2. Hence it is called a quadratic equation. (We know the quadratic equation is of the form $a{x^2} + bx + c = 0$, in our problem coefficient of ‘x’ is zero) Hence, we have two roots or two factors.