Structure of Expressions and Modeling with Equations: Algebra 2 Math Test Answer Key – PDF Download

Algebra 2 Expressions Test Answer Key: questions 1–5

Question 1: Interpret a model (2 points)

1. The factor 24000 is the starting value, $24,000. The factor 0.85 means the car keeps 85% of its value each year, a 15% yearly decrease.
2. \(V(3)=24000(0.85)^{3}=24000(0.614125)=14739\), so the value is $14,739.

Scoring: 1 point for both meanings (starting value and 15% decrease); 1 point for the correct value.

Question 2: Factor completely (2 points)

1. \(x^{4}-16y^{4}=(x^{2}+4y^{2})(x^{2}-4y^{2})\), so \((x^{2}+4y^{2})(x+2y)(x-2y)\).
2. \(x^{2}+49=x^{2}-(7i)^{2}\), so \((x+7i)(x-7i)\).

Scoring: 1 point for each correct factorization; part (a) must be factored completely.

Question 3: Binomial Theorem (3 points)

1. Row 4 is 1, 4, 6, 4, 1: \(x^{4}+4x^{3}(3)+6x^{2}(9)+4x(27)+81\) = \(x^{4}+12x^{3}+54x^{2}+108x+81\).
2. The \(x^{2}\) term uses the third power of \(-2\) and coefficient 10: \(10x^{2}(-2)^{3}=-80x^{2}\). The coefficient is −80.

Scoring: (a) 1 point for the Pascal coefficients, 1 point for the correct powers of 3; (b) 1 point.

Question 4: Rearrange formulas (2 points)

1. \(2\pi rh=S-2\pi r^{2}\), so \(h=\frac{S-2\pi r^{2}}{2\pi r}\).
2. \(y(x-1)=2x+5\), so \(xy-y=2x+5\) and \(x(y-2)=y+5\). Then \(x=\frac{y+5}{y-2}\).

Scoring: 1 point for each correct formula.

Question 5: Where two graphs meet (3 points)

1. The graphs cross at \(x=1\) and \(x=5\).
2. \(x^{2}-4x+1=2x-4\) gives \(x^{2}-6x+5=0\), so \((x-1)(x-5)=0\) and \(x=1\) or \(x=5\). Check: \(f(1)=g(1)=-2\) and \(f(5)=g(5)=6\).

Scoring: (a) 1 point for both values; (b) 1 point for the quadratic equation in standard form, 1 point for the factored solutions.

Algebra 2 Expressions Test Answer Key: questions 6–8

Question 6: Picture frame (3 points)

1. The outer rectangle is \(8+2x\) by \(10+2x\), so \((8+2x)(10+2x)=168\).
2. Expanding gives \(4x^{2}+36x+80=168\), so \(4x^{2}+36x-88=0\) and \(x^{2}+9x-22=0\). Then \((x+11)(x-2)=0\). The width cannot be negative, so reject \(x=-11\). The frame is 2 inches wide, and \(12\cdot 14=168\).

Scoring: (a) 1 point; (b) 1 point for the solved equation, 1 point for choosing 2 inches with the reason.

Question 7: Savings account growth (3 points)

1. \(A(t)=3500(1.06)^{t}\).
2. \(3500(1.06)^{t}=5000\).
3. \(A(6)=3500(1.06)^{6}\approx 4964.82\) is below 5000, and \(A(7)\approx 5262.71\) is above. The least whole number is 7 years.

Scoring: 1 point per part. Accept 1.06 written as \(1+0.06\).

Question 8: Constraints (2 points)

1. \(3n+5b\le 60\) and \(b\ge 4\).
2. \(3(10)+5(6)=60\le 60\) and \(6\ge 4\), so yes, the plan is possible. It lies on the boundary of the cost limit.

Scoring: (a) 1 point for both inequalities; (b) 1 point for checking both conditions.

Scoring summary

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