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Difficulty: MediumSimplifying Expressions and Combining Like Terms

For all real numbers uu and vv, which of the following is equivalent to the expression 12(2u−3v)2−23u(3u−9v)−23v2\frac{1}{2}(2u - 3v)^2 - \frac{2}{3}u(3u - 9v) - \frac{2}{3}v^2?

  1. 236v2\frac{23}{6}v^2Answer
  2. B
    6uv+236v26uv + \frac{23}{6}v^2
  3. C
    75v2\frac{7}{5}v^2
  4. D
    −12uv+236v2-12uv + \frac{23}{6}v^2
  5. E
    2u2−2u+236v22u^2 - 2u + \frac{23}{6}v^2

Answer

The simplified equivalent expression is 236v2\frac{23}{6}v^2.
The correct answer is obtained by expanding (2u−3v)2(2u - 3v)^2 to 4u2−12uv+9v24u^2 - 12uv + 9v^2, multiplying it by 12\frac{1}{2} to get 2u2−6uv+92v22u^2 - 6uv + \frac{9}{2}v^2, distributing −23u-\frac{2}{3}u to get −2u2+6uv-2u^2 + 6uv, and combining the terms: (2u2−2u2)+(−6uv+6uv)+(92−23)v2=236v2(2u^2 - 2u^2) + (-6uv + 6uv) + (\frac{9}{2} - \frac{2}{3})v^2 = \frac{23}{6}v^2.

Step-by-Step Solution

1
Expand the squared binomial (2u−3v)2(2u - 3v)^2.
4u2−12uv+9v24u^2 - 12uv + 9v^2
Using the binomial square formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 allows us to expand the expression before applying the outer coefficient.
2
Multiply the expanded binomial by the coefficient 12\frac{1}{2}.
2u2−6uv+92v22u^2 - 6uv + \frac{9}{2}v^2
Distributing the constant factor of 12\frac{1}{2} to each term of the expanded binomial.
3
Distribute the term −23u-\frac{2}{3}u across the parenthetical expression (3u−9v)(3u - 9v).
−2u2+6uv-2u^2 + 6uv
Multiplying each term inside the parentheses by −23u-\frac{2}{3}u, making sure to distribute the negative sign properly: −23u⋅3u=−2u2-\frac{2}{3}u \cdot 3u = -2u^2 and −23u⋅(−9v)=6uv-\frac{2}{3}u \cdot (-9v) = 6uv.
4
Combine all terms and group like terms.
2u2−6uv+92v2−2u2+6uv−23v22u^2 - 6uv + \frac{9}{2}v^2 - 2u^2 + 6uv - \frac{2}{3}v^2
Write the full expression with all distributed terms to identify and combine like terms.
5
Combine the u2u^2, uvuv, and v2v^2 terms.
236v2\frac{23}{6}v^2
Combining the coefficients: 2u2−2u2=02u^2 - 2u^2 = 0, −6uv+6uv=0-6uv + 6uv = 0, and 92v2−23v2=(276−46)v2=236v2\frac{9}{2}v^2 - \frac{2}{3}v^2 = (\frac{27}{6} - \frac{4}{6})v^2 = \frac{23}{6}v^2.

Key Concept

Simplifying expressions by expanding binomials, distributing negative signs, and combining like terms with fractional coefficients.

Alternative Method

Instead of algebraic expansion, you can substitute simple non-zero values for uu and vv (e.g., u=3u = 3 and v=2v = 2) into the original expression and evaluate it. Then, substitute the same values into the answer choices to find which one yields the same result.
Estimated Time:1m 30s
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Topics

Simplifying Expressions and Combining Like Terms

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