Answer
The solutions are $m = \frac{5}{2}$ and $m = -\frac{5}{2}$.
Work Step by Step
First, we can factor out what is common in the two terms:
$2(4m^2 - 25) = 0$
We can simplify the equation by dividing each side by $2$:
$4m^2 - 25 = 0$
We can see that this equation features the difference of two squares. To factor an equation in the form $a^2 - b^2 = 0$, we use the following formula:
$a^2 - b^2 = (a - b)(a + b)$
Let's apply this formula to our equation, where $a = m$ and $b = 5$:
$(m - 5)(m + 5) = 0$
According to the zero product property, if the product of two factors equals $0$, then either factor can be $0$; therefore, we can set each of these factors equal to $0$ and solve:
$2m - 5 = 0$ or $2m + 5 = 0$
Let's look at the first factor:
$2m - 5 = 0$
Add $5$ to both sides of the equation:
$2m = 5$
Divide each side by $2$:
$m = \frac{5}{2}$
Let's look at the second factor:
$2m + 5 = 0$
Subtract $5$ from both sides of the equation:
$2m = -5$
Divide each side by $2$:
$m = -\frac{5}{2}$
The solutions are $m = \frac{5}{2}$ and $m = -\frac{5}{2}$.
To check our answer, we substitute each of these values into the original equation to see if the two sides equal one another:
$8(\frac{5}{2})^2 - 50 = 0$
Evaluate exponents first:
$8(\frac{25}{4}) - 50 = 0$
Multiply:
$\frac{200}{4} - 50 = 0$
Simplify the fraction:
$50 - 50 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
Let's check the other solution:
$8(-\frac{5}{2})^2 - 50 = 0$
Evaluate exponents first:
$8(\frac{25}{4}) - 50 = 0$
Multiply:
$\frac{200}{4} - 50 = 0$
Simplify the fraction:
$50 - 50 = 0$
Subtract:
$0 = 0$
The two sides of the equation are equal to one another; therefore, this solution is correct.
