Question

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

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

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