A quadratic equation with roots -1$\frac{1}{2}$ and $\frac{1}{4}$ is

Analyze the question: We need a quadratic equation whose roots are -1½ (that is, -3/2) and 1/4.

Concept: If the roots are α and β, the equation is x² − (α + β)x + αβ = 0

Solution:

Sum of roots: -3/2 + 1/4 = -6/4 + 1/4 = -5/4

Product of roots: (-3/2)(1/4) = -3/8

Substitute into the formula: x² − (−5/4)x + (−3/8) = 0 x² + (5/4)x − 3/8 = 0

Multiply through by 8 to clear the fractions: 8x² + 10x − 3 = 0

Check with the options:

• Option B: 8x² + 10x + 3 = 0. The constant is +3, which is wrong.
• Option C: 8x² + 12x − 3 = 0. The middle term is wrong.
• Option D: 8x² + 10x − 3 = 0. This matches our result.

Quick verification: Put x = 1/4 into 8x² + 10x − 3: 8(1/16) + 10/4 − 3 = 0.5 + 2.5 − 3 = 0 ✓

Correct answer: D

Tip: Always use the pattern x² − (sum)x + (product) = 0. Watch the signs carefully: a negative sum becomes a positive middle term. Then multiply by the LCM of the denominators to get whole-number coefficients. Also, plugging one root back in is a quick way to confirm your answer.