Question

How do you factor the expression $25{x^2} - 64$?

Answer 1

We know that factorization of any expression or number means the splitting of expression or a number in such a way that factors are written in the product form and multiplying the factors will restore the original expression or number. So, to factorize the given expression we have to write the expression in the multiplication form. We can use the formula ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$ for this question.

Complete step-by-step answer:
The expression given in the above problem that we have to factorize is as follows:
$\Rightarrow 25{x^2} - 64$
Now, let’s solve the question.
If any number or variable is multiplied with itself a particular number of times, it defines its power. For example: If we multiply a variable ‘x’ 5 times, it will be $x \times x \times x \times x \times x$ which can be written as ${x^5}$. In the same way, if we multiply 4 four times, it will be $4 \times 4 \times 4 \times 4$ which can be written as ${4^4} = 256$. And if the power is 2 that means that a particular number or variable is multiplied two times. In the same way, if we see the question, here 64 is formed by multiplying 8 two times. So, it can be seen like this:
$\Rightarrow {\left( {5x} \right)^2} - {8^2}$
We know that ${a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)$.
Apply the above formula. That gives us
$\Rightarrow 25{x^2} - 64 = \left( {5x + 8} \right)\left( {5x - 8} \right)$

Hence, the factorized form of $25{x^2} - 64$ is $\left( {5x + 8} \right)\left( {5x - 8} \right)$.

Note:
Do remember all the identities of algebraic expressions. Students should know the square and square roots of the numbers at least 1 to 20. This will help factor the expression. By looking at the question, you should be able to identify which identity will fit that particular expression.
As we know some of the algebraic identities. Let’s discuss them.
${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$
${\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}$
$\left( {x + a} \right)\left( {x + b} \right) = {x^2} + \left( {a + b} \right)x + ab$