Question

How do you solve the equation by factoring and the zero-product rule: $4{x^2} - 25?$

Answer 1

Hint: Here we will solve the given expression by factorization using the formula for the difference of two squares and other by simplifying the terms for the unknown value by using the basic mathematics rules.

Complete step-by-step solution:
Method (I) by factorization:
Take the given expression equal to zero.
$\Rightarrow 4{x^2} - 25 = 0$
The above equation can be re-written as-
$\Rightarrow {(2x)^2} - {(5)^2} = 0$
Here we can apply the difference of two squares in the above equation. ${a^2} - {b^2} = (a - b)(a + b)$
$ \Rightarrow (2x - 5)(2x + 5) = 0$
We get two values-
$2x - 5 = 0$ and $2x + 5 = 0$
When you move any term from one side to another, then the sign of the term also changes. Positive terms become negative and the negative term becomes positive.
$2x = 5$ and $2x = ( - 5)$
Term multiplicative on one side if moved to the opposite side, then it goes to the denominator.
$ \Rightarrow x = \dfrac{5}{2}$ and $x = - \dfrac{5}{2}$

Method (II):
Take the given expression: $4{x^2} - 25 = 0$
Move the constant term on the opposite side of the equation. When you move any term to the opposite side, then its sign changes. Negative terms become positive.
$4{x^2} = 25$
Now the term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$ \Rightarrow {x^2} = \dfrac{{25}}{4}$
Take the square root on both the sides of the equation.
$ \Rightarrow \sqrt {{x^2}} = \sqrt {\dfrac{{25}}{4}} $
Square and square root cancel each other on the left hand side of the equation. Also the square root of positive numbers can give us a positive or negative square root.
$ \Rightarrow x = \sqrt {{{\left( { \pm \dfrac{5}{2}} \right)}^2}} $
Square and square root cancel each other.
$ \Rightarrow x = \pm \dfrac{5}{2}$
This is the required solution.

Note: Be careful about the sign convention while simplification of the equations. Always remember the square of positive number and the negative number always gives the resultant value as the positive term but square root of positive number can be positive or the negative term.