Question

How do you factor $9{x^2} - 4$?

Answer 1

Hint: As we know that factorising is the reverse of expanding brackets, it is an important way of solving equations. The first step of factoring an expression is to take out any common factors which the terms have. So if we were asked to factor the expression ${x^2} + x$, since $x$ goes into both terms, we would write $x(x + 1)$. Here we will use identities which will help us to factorise an algebraic expression easily i.e. ${(a - b)^2} = {a^2} - 2ab + {b^2}$ and ${a^2} - {b^2} = (a + b)(a - b)$.

Complete step by step answer:
Here we will use some identities to help like the difference of square identity:
${a^2} - {b^2} = (a + b)(a - b)$
We can further write $9{x^2} - 4 = {(3x)^2} - {(2)^2}$, Let ${a^2} = 9{x^2}$and ${b^2} = 4$ then we have $a = 3x$ and $b = 4$.
On further simplifying the above expression by applying the difference formula we get:
$9{x^2} - 4 = (3x + 2)(3x - 2)$.
Hence the factor of the equation $9{x^2} - 4$ is $(3x + 2)(3x - 2)$.

Note: We should keep in mind while solving these expressions that we use correct identities to factorise the given algebraic expressions and keep checking the negative and positive sign otherwise it will give the wrong answer. Also we should know that the difference of the square formula is the identity that is used in the above solution. These are some of the standard algebraic identities. This is as far we can go with real coefficients as the remaining quadratic factors all have complex zeros.