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

Question

(23+m4)243(m423)2=? (\frac{2}{3}+\frac{m}{4})^2-\frac{4}{3}-(\frac{m}{4}-\frac{2}{3})^2=\text{?}

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:

  • First, expand both squares:
    • (23+m4)2=(23)2+2×23×4×m+(m4)2=49+m6+m216 \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}
    • (m423)2=(m4)22×m12+(23)2=m216m6+49 \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}
  • Now substitute and simplify into the expression:
    • Expression becomes: (49+m6+m216)43(m216m6+49)\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 m216m216 \frac{m^2}{16} - \frac{m^2}{16} cancels out.
    • Now simplify the remaining terms: 49+m643+m649=2m643 \frac{4}{9} + \frac{m}{6} - \frac{4}{3} + \frac{m}{6} - \frac{4}{9} = \frac{2m}{6} - \frac{4}{3}
  • Use common denominators to combine final terms:
    • 2m6=m3\frac{2m}{6} = \frac{m}{3} , 43=129 \frac{4}{3} = \frac{12}{9} , resulting in: m3129 \frac{m}{3} - \frac{12}{9}
  • Recognize that this is a difference of squares:
    • m3129=(2m+2)(2m2)3 \frac{m}{3} - \frac{12}{9} = \frac{(\sqrt{2m} + 2)(\sqrt{2m} - 2)}{3}

Therefore, the simplified expression is given by the choice: (2m+2)(2m2)3 \frac{(\sqrt{2m}+2)(\sqrt{2m}-2)}{3} .

Answer

(2m+2)(2m2)3 \frac{(\sqrt{2m}+2)(\sqrt{2m}-2)}{3}