Question 1

If the roots of the equation 2x² - 8x + k = 0 are equal, find the value of k.

Accepted Answer

Step 1: For equal roots, the discriminant Δ = b² - 4ac = 0. Step 2: Here, a = 2, b = -8, c = k. Step 3: (-8)² - 4(2)(k) = 0. Step 4: 64 - 8k = 0. Step 5: 8k = 64. Step 6: k = 8. Therefore, the answer is 8.

Question 2

If the roots of x² - 6x + m = 0 are equal, find the value of m.

Accepted Answer

Step 1: For equal roots, the discriminant must be zero: b² - 4ac = 0. Step 2: Here a = 1, b = -6, c = m. Step 3: (-6)² - 4(1)(m) = 0. Step 4: 36 - 4m = 0. Step 5: 4m = 36, so m = 9. Therefore, the answer is 9.

Question 3

Solve: x² - 9 = 0. What are the roots?

Accepted Answer

Step 1: Recognize this as a difference of squares: x² - 9 = (x - 3)(x + 3) = 0. Step 2: Set each factor to zero: x - 3 = 0 or x + 3 = 0. Step 3: x = 3 or x = -3. Therefore, the roots are 3 and -3.

Question 4

If one root of the equation x² - 5x + k = 0 is 2, find the value of k.

Accepted Answer

Step 1: If x = 2 is a root, it must satisfy the equation. Step 2: Substitute x = 2: (2)² - 5(2) + k = 0. Step 3: 4 - 10 + k = 0. Step 4: -6 + k = 0, so k = 6. Therefore, the answer is 6.

Question 5

What is the product of the roots of the equation 2x² - 8x + 6 = 0?

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Here a = 2, b = -8, c = 6. Step 3: Product of roots = 6/2 = 3. Therefore, the answer is 3.

Question 6

If x² - 7x + 12 = 0, find the sum of the roots of the 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, the answer is 7.

Question 7

What is the product of the roots of the equation 3x² + 6x - 9 = 0?

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Here, a = 3, b = 6, c = -9. Step 3: Product of roots = -9/3 = -3. Therefore, the answer is -3.

Question 8

What is the product of the roots of the equation 2x² - 8x + 6 = 0?

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Here a = 2, b = -8, c = 6. Step 3: Product of roots = 6/2 = 3. Therefore, the answer is 3.

Question 9

Solve: x² - 5x + 6 = 0. What are the roots?

Accepted Answer

Step 1: Factor the quadratic: x² - 5x + 6 = (x - 2)(x - 3) = 0. Step 2: Set each factor to zero: x - 2 = 0 or x - 3 = 0. Step 3: Roots are x = 2 and x = 3. Therefore, the answer is 2 and 3.

Question 10

If one root of the equation x² - 6x + k = 0 is 4, find the value of k.

Accepted Answer

Step 1: If x = 4 is a root, substitute it into the equation: (4)² - 6(4) + k = 0. Step 2: Simplify: 16 - 24 + k = 0. Step 3: -8 + k = 0, so k = 8. Therefore, the answer is 8.

Question 11

The quadratic equation x² + 4x + 4 = 0 has roots. What is the nature of these roots?

Accepted Answer

Step 1: Calculate the discriminant: Δ = b² - 4ac. Step 2: Here a = 1, b = 4, c = 4. Step 3: Δ = (4)² - 4(1)(4) = 16 - 16 = 0. Step 4: When Δ = 0, the equation has equal (repeated) real roots. Therefore, the answer is equal real roots.

Question 12

If the sum of roots of the equation 3x² + (2m - 1)x + 5 = 0 is -2, find the value of m.

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, sum of roots = -b/a. Step 2: Here a = 3, b = (2m - 1), and sum of roots = -2. Step 3: Set up equation: -(2m - 1)/3 = -2. Step 4: -(2m - 1) = -6. Step 5: 2m - 1 = 6. Step 6: 2m = 7, so m = 7/2 = 3.5. Therefore, the answer is 3.5.

Question 13

If one root of the equation x² - 5x + 6 = 0 is 2, what is the other root?

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the sum of roots = -b/a. Step 2: Here, a = 1, b = -5, so sum of roots = 5. Step 3: If one root is 2, then the other root = 5 - 2 = 3. Alternatively, using product of roots: product = c/a = 6/1 = 6. If one root is 2, then other root = 6/2 = 3. Therefore, the answer is 3.

Question 14

If x² - 7x + 12 = 0, find the sum of the roots of the 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, the answer is 7.

Question 15

If the roots of the quadratic equation x² − 5x + k = 0 are in the ratio 2:3, find the value of k.

Accepted Answer

Step 1: Let the roots be 2α and 3α. Step 2: By Vieta's formulas, sum of roots = 2α + 3α = 5α = 5, so α = 1. Step 3: The roots are 2 and 3. Step 4: Product of roots = 2 × 3 = 6 = k. Therefore, k = 6.

Question 16

If the roots of the quadratic equation x² - 10x + k = 0 are in the ratio 2:3, find the value of k.

Accepted Answer

Step 1: Let the roots be 2α and 3α. Step 2: Sum of roots = 2α + 3α = 5α = 10, so α = 2. Step 3: The roots are 4 and 6. Step 4: Product of roots = 4 × 6 = 24 = k. Therefore, k = 24.

Question 17

The roots of the equation 2x² - 8x + k = 0 are reciprocals of each other. Find k.

Accepted Answer

Step 1: Let the roots be α and 1/α (reciprocals). Step 2: By Vieta's formulas, product of roots = k/2. Step 3: Since the roots are reciprocals: α × (1/α) = 1. Step 4: Therefore, k/2 = 1, which gives k = 2. Step 5: Verify: 2x² - 8x + 2 = 0 simplifies to x² - 4x + 1 = 0. Using the quadratic formula: x = (4 ± √(16-4))/2 = (4 ± √12)/2 = 2 ± √3. Product = (2+√3)(2-√3) = 4 - 3 = 1. ✓

Question 18

The sum of the squares of two consecutive integers is 145. Which quadratic equation represents this condition?

Accepted Answer

Step 1: Let the two consecutive integers be n and n+1. Step 2: Sum of squares: n² + (n+1)² = 145. Step 3: Expand: n² + n² + 2n + 1 = 145. Step 4: Simplify: 2n² + 2n + 1 = 145. Step 5: Rearrange: 2n² + 2n - 144 = 0. Step 6: Divide by 2: n² + n - 72 = 0. Therefore, the equation is n² + n - 72 = 0.

Question 19

If α and β are roots of x² - 5x + 6 = 0, find the value of (α - β)².

Accepted Answer

Step 1: For equation x² - 5x + 6 = 0, sum of roots α + β = 5 and product αβ = 6. Step 2: We know (α - β)² = (α + β)² - 4αβ. Step 3: (α - β)² = 5² - 4(6) = 25 - 24 = 1. Therefore, (α - β)² = 1.

Question 20

A quadratic equation has roots that differ by 5. If the sum of the roots is 7, what is the equation?

Accepted Answer

Step 1: Let the roots be α and β. Given: α − β = 5 and α + β = 7. Step 2: Solve: Adding, 2α = 12, so α = 6. Subtracting, 2β = 2, so β = 1. Step 3: Sum of roots = 6 + 1 = 7 ✓ and difference = 6 − 1 = 5 ✓. Step 4: Product of roots = 6 × 1 = 6. Step 5: Equation: x² − (sum)x + (product) = 0 → x² − 7x + 6 = 0. Therefore, the equation is x² − 7x + 6 = 0.

RRB NTPC Quadratic Equations

Quadratic Equations— Rules & Concept

Key Points to Remember

Exam-Specific Tips

Quadratic Equations — Practice Questions

60-Second Revision — Quadratic Equations