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How do you write \[9{x^2} - 4\] in factored form?

seo-qna
Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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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 \[9{x^2} - 4\].

Additional Information:
We can find the root of the polynomial by equating the obtained factors to zero. That is
\[ \Rightarrow (3x - 2)(3x + 2) = 0\]
Using the zero product principle, that is of ab=0 then a=0 or b=0.using this we get,
\[ \Rightarrow 3x - 2 = 0\] 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.
We know that on the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x- intercept.

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.