Question 1

If α and β are the roots of the equation 2x² − 7x + 3 = 0, find the value of α³ + β³.

Accepted Answer

Step 1: By Vieta's formulas, α + β = 7/2 and αβ = 3/2. Step 2: α² + β² = (α + β)² − 2αβ = (49/4) − 3 = 37/4. Step 3: α³ + β³ = (α + β)(α² − αβ + β²) = (α + β)[(α² + β²) − αβ] = (7/2)[(37/4) − (3/2)] = (7/2)(37/4 − 6/4) = (7/2)(31/4) = 217/8. Therefore, α³ + β³ = 217/8.

Question 2

If the roots of x² + px + 12 = 0 are 3 and 4, find the value of p.

Accepted Answer

Step 1: For a quadratic equation, sum of roots = -p/1 = -p. Step 2: Sum of roots = 3 + 4 = 7. Step 3: Therefore, -p = 7, which gives p = -7. Therefore, the answer is -7.

Question 3

If the roots of the equation x² - 5x + k = 0 are 2 and 3, find the value of k.

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Product of roots = 2 × 3 = 6. Step 3: Here, a = 1 and c = k, so k/1 = 6. Step 4: Therefore, k = 6.

Question 4

Solve: 2x² - 8x = 0. What is the product of the roots?

Accepted Answer

Step 1: Factor the equation: 2x(x - 4) = 0. Step 2: The roots are x = 0 and x = 4. Step 3: Product of roots = 0 × 4 = 0. Alternatively, using Vieta's formula: product = c/a = 0/2 = 0. Therefore, the answer is 0.

Question 5

If one root of the equation x² - 6x + 8 = 0 is 2, find the other root.

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: Here, a = 1, b = -6, c = 8, so product of roots = 8/1 = 8. Step 3: If one root is 2, then 2 × (other root) = 8. Step 4: Other root = 8/2 = 4. Therefore, the answer is 4.

Question 6

The quadratic equation x² - 4x + 3 = 0 has roots α and β. Find the value of α² + β².

Accepted Answer

Step 1: Sum of roots (α + β) = 4 and product of roots (αβ) = 3. Step 2: Use the identity α² + β² = (α + β)² - 2αβ. Step 3: α² + β² = (4)² - 2(3) = 16 - 6 = 10. Therefore, the answer is 10.

Question 7

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 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

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, then 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 10

The quadratic equation x² - (a+b)x + ab = 0 has roots that differ by 5. If one root is 3, find the value of |a - b|.

Accepted Answer

Step 1: The equation x² - (a+b)x + ab = 0 has roots a and b (by inspection/factorization: (x-a)(x-b) = 0). Step 2: One root is 3, so either a = 3 or b = 3. Step 3: The roots differ by 5, so |a - b| = 5. Step 4: If a = 3 and |a - b| = 5, then b = -2 or b = 8. Step 5: Check: if a = 3 and b = -2, roots are 3 and -2, difference = 5 ✓. If a = 3 and b = 8, roots are 3 and 8, difference = 5 ✓. Step 6: In either case, |a - b| = 5.

Question 11

A quadratic equation x² + px + q = 0 has roots that differ by 5. If the sum of the roots is 7, find the value of q.

Accepted Answer

Step 1: Let the roots be α and β. Given: α + β = 7 and |α - β| = 5. Step 2: From Vieta's formula, α + β = -p = 7, so p = -7. Step 3: We have two equations: α + β = 7 and α - β = 5 (assuming α > β). Step 4: Adding: 2α = 12 → α = 6. Step 5: Subtracting: 2β = 2 → β = 1. Step 6: Product of roots q = αβ = 6 × 1 = 6.

Question 12

A quadratic equation x² - 8x + k = 0 has two real roots. If the difference between the roots is 4, find the value of k.

Accepted Answer

Step 1: Let the roots be α and β. Step 2: Sum of roots: α + β = 8. Step 3: Difference of roots: |α - β| = 4. Step 4: From these two equations, assume α > β, so α - β = 4. Step 5: Adding: (α + β) + (α - β) = 8 + 4 → 2α = 12 → α = 6. Step 6: Subtracting: (α + β) - (α - β) = 8 - 4 → 2β = 4 → β = 2. Step 7: Product of roots: k = αβ = 6 × 2 = 12.

Question 13

If the sum of the roots of the quadratic equation 2x² + (k - 3)x + 5 = 0 is equal to the product of the roots, find the value of k.

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, sum of roots = -b/a and product of roots = c/a. Step 2: Here, a = 2, b = (k - 3), c = 5. Step 3: Sum of roots = -(k - 3)/2 = (3 - k)/2. Step 4: Product of roots = 5/2. Step 5: Setting them equal: (3 - k)/2 = 5/2. Step 6: 3 - k = 5, so k = -2.

Question 14

The sum of the squares of two consecutive integers is 145. Find the larger integer.

Accepted Answer

Step 1: Let the two consecutive integers be n and n+1. Step 2: Set up equation: n² + (n+1)² = 145. Step 3: Expand: n² + n² + 2n + 1 = 145. Step 4: Simplify: 2n² + 2n + 1 = 145. Step 5: 2n² + 2n - 144 = 0. Step 6: Divide by 2: n² + n - 72 = 0. Step 7: Factor: (n + 9)(n - 8) = 0. Step 8: n = -9 or n = 8. Step 9: If n = 8, the integers are 8 and 9. Check: 64 + 81 = 145 ✓. Step 10: The larger integer is 9.

Question 15

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

Accepted Answer

Step 1: Since x = 2 is a root, substitute into the equation: 2(2)² - 5(2) + c = 0. Step 2: 8 - 10 + c = 0. Step 3: c = 2. Step 4: The equation is 2x² - 5x + 2 = 0. Step 5: Using Vieta's formula, product of roots = c/a = 2/2 = 1. Step 6: If one root is 2, then 2 × (other root) = 1. Step 7: Other root = 1/2.

Question 16

The roots of the equation x² - 7x + 12 = 0 are α and β. Find the value of (α + β)² - 2αβ.

Accepted Answer

Step 1: For x² - 7x + 12 = 0, by Vieta's formulas: α + β = 7 and αβ = 12. Step 2: Calculate (α + β)² = 7² = 49. Step 3: Calculate 2αβ = 2 × 12 = 24. Step 4: (α + β)² - 2αβ = 49 - 24 = 25.

Question 17

If α and β are roots of x² - 6x + 5 = 0, and γ and δ are roots of x² - 3x + 2 = 0, then the value of (α - γ)(α - δ)(β - γ)(β - δ) is:

Accepted Answer

Step 1: For x² - 6x + 5 = 0, factoring gives (x - 1)(x - 5) = 0, so α = 1, β = 5 (or vice versa). Step 2: For x² - 3x + 2 = 0, factoring gives (x - 1)(x - 2) = 0, so γ = 1, δ = 2 (or vice versa). Step 3: Calculate (α - γ)(α - δ)(β - γ)(β - δ) = (1 - 1)(1 - 2)(5 - 1)(5 - 2) = 0 × (-1) × 4 × 3 = 0. Step 4: Alternatively, note that α = 1 is a common root, so (α - γ) = 0 when γ = 1, making the entire product 0.

Question 18

A quadratic equation x² - 6x + k = 0 has two distinct real roots. If one root is three times the other, find the value of k.

Accepted Answer

Step 1: Let the roots be r and 3r. Step 2: By Vieta's formulas, sum of roots: r + 3r = 6, so 4r = 6, giving r = 1.5. Step 3: Product of roots: r × 3r = k, so 3r² = k. Step 4: Substitute r = 1.5: k = 3 × (1.5)² = 3 × 2.25 = 6.75.

Question 19

If the roots of the quadratic equation x² + px + q = 0 are α and β, and the roots of x² + mx + n = 0 are α² and β², then which of the following is true?

Accepted Answer

Step 1: For equation x² + px + q = 0, by Vieta's formulas: α + β = -p and αβ = q. Step 2: For equation x² + mx + n = 0, the sum of roots is α² + β² = (α + β)² - 2αβ = p² - 2q, so m = -(p² - 2q) = 2q - p². Step 3: The product of roots is α²β² = (αβ)² = q², so n = q². Step 4: Therefore, m = 2q - p² and n = q².

Question 20

The sum of the squares of the roots of the equation x² - 5x + 3 = 0 is equal to:

Accepted Answer

Step 1: For equation x² - 5x + 3 = 0, let roots be α and β. By Vieta's formulas: α + β = 5 and αβ = 3. Step 2: We need α² + β². Step 3: Use the identity (α + β)² = α² + 2αβ + β², so α² + β² = (α + β)² - 2αβ. Step 4: α² + β² = 5² - 2(3) = 25 - 6 = 19.

Question 21

If α and β are roots of x² + 4x + 2 = 0, then the value of (1/α) + (1/β) is:

Accepted Answer

Step 1: For equation x² + 4x + 2 = 0, by Vieta's formulas: α + β = -4 and αβ = 2. Step 2: We need (1/α) + (1/β) = (β + α)/(αβ). Step 3: Substitute values: (1/α) + (1/β) = (-4)/2 = -2.

SSC CHSL Quadratic Equations

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Quadratic Equations — Practice Questions

60-Second Revision — Quadratic Equations