Simplify the Expression: Solving (2/3 + m/4)² - 4/3 - (m/4 - 2/3)²

Video Solution

Solution Steps

00:00 Simply

00:03 Let's use shortened multiplication formulas to open the parentheses

00:21 Square both numerator and denominator

00:26 Make sure to multiply numerator by numerator and denominator by denominator

00:31 Square both numerator and denominator

00:35 Reduce what's possible

00:41 Let's substitute in our exercise

00:58 Let's use shortened multiplication formulas to open the parentheses

01:16 Square both numerator and denominator

01:21 Make sure to multiply numerator by numerator and denominator by denominator

01:25 Reduce what's possible

01:37 Let's substitute in our exercise

01:50 Negative times positive is always negative

01:54 Negative times negative is always positive

02:02 Collect like terms

02:29 Let's use shortened multiplication formulas and write in a different form

02:32 And this is the solution to the question

Step-by-Step Solution

To solve this problem, let's follow a detailed approach:

$\left(\frac{2}{3} + \frac{m}{4}\right)^2 = \left(\frac{2}{3}\right)^2 + \frac{2 \times 2}{3 \times 4} \times m + \left(\frac{m}{4}\right)^2 = \frac{4}{9} + \frac{m}{6} + \frac{m^2}{16}$
$\left(\frac{m}{4} - \frac{2}{3}\right)^2 = \left(\frac{m}{4}\right)^2 - \frac{2 \times m}{12} + \left(\frac{2}{3}\right)^2 = \frac{m^2}{16} - \frac{m}{6} + \frac{4}{9}$

Expression becomes: $\left( \frac{4}{9} + \frac{m}{6} + \frac{m^2}{16} \right) - \frac{4}{3} - \left( \frac{m^2}{16} - \frac{m}{6} + \frac{4}{9} \right)$
Observe that $\frac{m^2}{16} - \frac{m^2}{16}$ cancels out.
Now simplify the remaining terms: $\frac{4}{9} + \frac{m}{6} - \frac{4}{3} + \frac{m}{6} - \frac{4}{9} = \frac{2m}{6} - \frac{4}{3}$

$\frac{2m}{6} = \frac{m}{3}$ , $\frac{4}{3} = \frac{12}{9}$ , resulting in: $\frac{m}{3} - \frac{12}{9}$

Therefore, the simplified expression is given by the choice: $\frac{(\sqrt{2m}+2)(\sqrt{2m}-2)}{3}$ .