Question

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

Answer 1

Hint: In this question we will try to simplify the expression and then use the formula of expansion. After doing some simplification we get the required answer.

Formula used: ${a^2} - {b^2} = (a - b)(a + b)$

Complete step-by-step solution:
We have the given expression as: $25{x^2} - 64$
Now we can see in the expression that the terms $25$ and $64$ are present, which have square roots therefore, we can utilize their square roots to simplify the expression.
We know that ${5^2} = 25$ and ${8^2} = 64$ therefore, on substituting it in the expression, we get:
$ \Rightarrow {5^2}{x^2} - {8^2}$
Now we know the property of exponent that ${a^n}{b^n} = a{b^n}$ therefore the expression can be written as:
$ \Rightarrow {(5x)^2} - {8^2}$
Now the above expression is in the form of ${a^2} - {b^2}$ therefore, we can expand it using the expansion formula.
On using the formula, we get:
$ \Rightarrow (5x - 8)(5x + 8)$

$(5x-8)(5x+8)$ is the required solution.

Note: To check whether the solution is correct, we will multiply the factors, if it gives the same term, then the solution is correct:
On multiplying, we get:
$ \Rightarrow 5x \times 5x + 5x \times 8 - 8 \times 5x - 8 \times 8$
On simplifying, we get:
$ \Rightarrow 25{x^2} + 40x - 40x - 64$
On cancelling the terms, we get:
$ \Rightarrow 25{x^2} - 64$, which is the original expression therefore, the solution is correct.
In the above question we have a polynomial equation of degree$2$. Even though the polynomial equation has a degree $2$, it is not a complete quadratic equation because it has the $x$ coefficient as $0$.
The various expansion formulae should be remembered such as ${(a + b)^2}$ and ${(a - b)^2}$ to do these types of questions.
It is to be remembered that factors are the digits which make up a number, a factor should be indivisible by any number except $1$ and it should not be in the form of a number which is raised to a power.