Question

How do you factor \[36{x^2} - 49\] ?

Answer 1

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 7 is 49 and square of 36 is 6.

Complete step by step answer:
Given, \[36{x^2} - 49\]
We can rewrite it as \[ = {6^2}.{x^2} - {7^2}\]
\[ 36{x^2} - 49 = {(6x)^2} - {7^2}\].
That is it is in the form \[{a^2} - {b^2}\], where \[a = 6x\] and \[b = 7\].
We have the formula \[{a^2} - {b^2} = (a - b)(a + b)\].
Then above becomes,
\[36{x^2} - 49 = (6x - 7)(6x + 7)\]. These are the factors of the \[36{x^2} - 49\].

Additional Information:
We can find the root of the polynomial by equating the obtained factors to zero. That is
\[(6x - 7)(6x + 7) = 0\]
Using the zero product principle, that is of ab=0 then a=0 or b=0.Using this we get,
\[6x - 7 = 0\] and \[6x + 7 = 0\].
\[ \Rightarrow 6x = 7\] and \[6x = - 7\].
\[ \Rightarrow x = \dfrac{7}{6}\] and \[x = - \dfrac{7}{6}\]
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.