Structure of Expressions and Modeling Test Answer Key with Scoring – Algebra 2

Question 1 Factor completely (2 points)

(x² + 16)(x² − 16) = (x² + 16)(x + 4)(x − 4).

Scoring: 1 point for the first difference of squares, 1 point for the complete factoring.

Question 2 Rearrange formulas (2 points)

1. 9C/5 = F − 32, so F = 9C/5 + 32.
2. v² − u² = 2as, so a = (v² − u²)/(2s).

Scoring: 1 point each.

Question 3 Binomial expansion (3 points)

Row 4: 1, 4, 6, 4, 1. x⁴ + 4x³(−2) + 6x²(4) + 4x(−8) + 16 = x⁴ − 8x³ + 24x² − 32x + 16.

Scoring: 1 point for the coefficients, 1 point for the powers of −2, 1 point for the final answer.

Question 4 Complex factoring (2 points)

1. (x + 7i)(x − 7i).
2. x = (6 ± √(36 − 100))/2 = (6 ± 8i)/2 = 3 ± 4i.

Scoring: 1 point each.

Question 5 Intersection of two graphs (3 points)

1. The graphs cross near x = −2 and x = 3.
2. x² − 4 = x + 2 gives x² − x − 6 = 0, so (x − 3)(x + 2) = 0. x = 3 or x = −2, at the points (3, 5) and (−2, 0).

Scoring: 1 point for (a), 1 point for the equation, 1 point for both solutions.

Question 6 Scooter depreciation (3 points)

1. V(t) = 1800(0.85)^t.
2. 1800(0.85)³ = 1800(0.614125) = $1,105.43.
3. 0.85^t = 0.5, so t = ln 0.5 ÷ ln 0.85 = 4.265, or ≈ 4.3 years.

Scoring: 1 point per part.

Question 7 Ticket system (2 points)

x + y = 120 and 9x + 5y = 780. Substitute: 9x + 600 − 5x = 780, so 4x = 180 and x = 45. 45 adult and 75 child tickets. Check: 405 + 375 = 780.

Scoring: 1 point for the equations, 1 point for the solution.

Question 8 Pen with a minimum area (2 points)

Length 20 − w, so w(20 − w) ≥ 91. Then w² − 20w + 91 ≤ 0, so (w − 7)(w − 13) ≤ 0. 7 ≤ w ≤ 13 feet.

Scoring: 1 point for the inequality, 1 point for the interval.

Question 9 Check an identity (1 point)

Left: (25 + 4)² = 841. Right: (25 − 4)² + (20)² = 441 + 400 = 841. Both sides equal 841.

Scoring: 1 point for both sides.