Directions: Solve the quadratic equations and answer the question that…

Concept

A quadratic equation ax2 + bx + c = 0 has two roots. To find an unknown constant when one root is known, substitute that root into the equation and solve for the constant; then factor or use the quadratic formula to obtain both roots. The sum of the roots equals -b/a and the product equals c/a, which gives an independent check.

Application

1. Bring equation A to standard form. Expand (x-2)2 = x2 - 4x + 4 and move every term to the left of x2 - 4x + 4 = -3x2 + 4 + 25x - P.

2. Collecting like terms gives 4x2 - 29x + P = 0.

3. One root is given as 5, so substitute x = 5: 4(25) - 29(5) + P = 100 - 145 + P = 0, hence P = 45.

4. The equation becomes 4x2 - 29x + 45 = 0. Factor: 4x2 - 20x - 9x + 45 = 4x(x - 5) - 9(x - 5) = (x - 5)(4x - 9) = 0.

5. The roots are x = 5 and x = 9/4 = 2.25. The smaller root is 9/4.

6. The largest one-digit prime number is 7 (the primes below 10 are 2, 3, 5, 7). Multiply the smaller root by it: (9/4) × 7 = 63/4 = 15 3/4.

Cross-check

Verify with sum and product of roots of 4x2 - 29x + 45 = 0: sum = 29/4 = 5 + 9/4 and product = 45/4 = 5 × 9/4, both consistent, so the roots 5 and 9/4 are correct. Therefore the required value is 63/4 = 15 3/4.