Directions: Solve the quadratic and answer the following question. A: (x − 2)2…

Concept

If a number r is a root of a quadratic equation, then substituting x = r makes the equation true. So we rewrite the equation in the standard form ax2 + bx + c = 0, plug in the known root, and solve the resulting linear equation for the unknown constant.

Application

1. Expand and rearrange equation A. (x − 2)2 = −3x2 + 22 + 25x − P becomes x2 − 4x + 4 = −3x2 + 4 + 25x − P.

2. Move every term to the left: x2 − 4x + 4 + 3x2 − 4 − 25x + P = 0, i.e. 4x2 − 29x + P = 0.

3. Use the given root x = 5: 4(5)2 − 29(5) + P = 0 → 100 − 145 + P = 0 → P = 45.

4. Substitute P = 45 into the target expression (7P / 21) × 0.2P − 89.

5. (7 × 45 / 21) = 15 and 0.2 × 45 = 9, so 15 × 9 = 135; then 135 − 89 = 46.

Cross-check

With P = 45, the four algebraic options evaluate to P+2 = 47, 2P+2 = 92, P+1 = 46 and 2P−1 = 89. Only P+1 equals the computed value 46, confirming the result.