Question 1

If x² − 7x + 12 = 0, find the sum of the roots of this quadratic equation.

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the sum of roots = −b/a. Step 2: Here, a = 1, b = −7, c = 12. Step 3: Sum of roots = −(−7)/1 = 7. Therefore, answer is B.

Question 2

If the roots of the quadratic equation x² − 5x + k = 0 are α and β, and α² + β² = 13, find the value of k.

Accepted Answer

Step 1: Use the relationship between roots and coefficients. For x² − 5x + k = 0, we have α + β = 5 and αβ = k. Step 2: Express α² + β² in terms of sum and product: α² + β² = (α + β)² − 2αβ = 5² − 2k = 25 − 2k. Step 3: Set this equal to 13: 25 − 2k = 13. Step 4: Solve for k: 2k = 12, so k = 6. Therefore, answer is B.

Question 3

If the roots of the quadratic equation x² − (p + q)x + pq = 0 are α and β, and the roots of the equation x² − (α + β)x + αβ = 0 are m and n, then what is the relationship between the sum of roots (m + n) and the product of roots (mn) of the second equation?

Accepted Answer

Step 1: For the first equation x² − (p + q)x + pq = 0, by Vieta's formulas, the roots α and β satisfy: α + β = p + q and αβ = pq. Step 2: The second equation is x² − (α + β)x + αβ = 0, which becomes x² − (p + q)x + pq = 0. Step 3: This is identical to the first equation. Step 4: Therefore, the roots m and n of the second equation are the same as α and β. Step 5: By Vieta's formulas for the second equation: m + n = α + β = p + q, and mn = αβ = pq. Step 6: The relationship is that m + n = p + q and mn = pq, meaning (m + n) and (mn) equal the sum and product of the original roots respectively. Therefore, answer is B.

RRB Group D Quadratic Equations

Quadratic Equations— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Quadratic Equations — Practice Questions

60-Second Revision — Quadratic Equations