Question 1

Find the roots of the quadratic equation x² - 5x + 6 = 0.

Accepted Answer

Step 1: Factorise the equation. We need two numbers whose product is 6 and sum is 5. Those numbers are 2 and 3. Step 2: Write as (x - 2)(x - 3) = 0. Step 3: Setting each factor to zero gives x = 2 or x = 3. Therefore, the roots are 2 and 3.

Question 2

If the roots of the quadratic equation x² − (p + q)x + pq = 0 are α and β, and α − β = 6, then which of the following is true?

Accepted Answer

Step 1: For the equation x² − (p + q)x + pq = 0, by Vieta's formulas: α + β = p + q and αβ = pq. Step 2: We are given α − β = 6. Step 3: Using the identity (α − β)² = (α + β)² − 4αβ, we get: 36 = (p + q)² − 4pq. Step 4: Expanding: 36 = p² + 2pq + q² − 4pq = p² − 2pq + q² = (p − q)². Step 5: Therefore, p − q = ±6. The relationship between p and q is that their difference equals ±6.

Question 3

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

Accepted Answer

Step 1: For a quadratic equation ax² + bx + c = 0, 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 other root = 5 - 2 = 3. Therefore, the answer is 3.

Question 4

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

Accepted Answer

Step 1: If roots are 3 and 4, then sum of roots = 3 + 4 = 7. Step 2: For the equation x² + px + 12 = 0, sum of roots = -p/1 = -p. Step 3: Therefore, -p = 7, which gives p = -7. Therefore, the answer is -7.

Question 5

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 6

If the roots of x² + bx + 20 = 0 are 4 and 5, find the value of b.

Accepted Answer

Step 1: Use Vieta's formula: sum of roots = -b/a. Step 2: Here a = 1, and the roots are 4 and 5. Step 3: Sum of roots = 4 + 5 = 9. Step 4: 9 = -b/1, so b = -9. Therefore, the answer is -9.

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

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.

Question 9

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 must equal zero: 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: k = 8. Therefore, the answer is 8.

Question 10

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 11

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: Therefore x = 2 or x = 3. The roots are 2 and 3.

Question 12

Which of the following is a root of x² - 6x + 8 = 0?

Accepted Answer

Step 1: Factor the quadratic: x² - 6x + 8 = (x - 2)(x - 4) = 0. Step 2: Set each factor to zero: x - 2 = 0 or x - 4 = 0. Step 3: x = 2 or x = 4. Step 4: Both 2 and 4 are roots. Checking the options, 4 is listed. Therefore, the answer is 4.

Question 13

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 14

The product of the roots of the equation 3x² + 6x - 9 = 0 is:

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 15

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 16

Find 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 17

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 two equal real roots (or one repeated root). Therefore, the roots are equal and real.

Question 18

If one root of the equation x² - 6x + 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 - 6(2) + k = 0. Step 3: 4 - 12 + k = 0. Step 4: -8 + k = 0, so k = 8. Therefore, k = 8.

Question 19

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 20

If the roots of the quadratic equation x² − (p + q)x + pq = 0 are α and β, and α − β = 6, then find the value of (α + β)² − 4αβ in terms of the roots.

Accepted Answer

Step 1: By Vieta's formulas, α + β = p + q and αβ = pq. Step 2: We know (α − β)² = (α + β)² − 4αβ. Step 3: Given α − β = 6, so (α − β)² = 36. Step 4: Therefore, (α + β)² − 4αβ = 36.

Question 21

If α and β are roots of x² - 6x + k = 0, and α² + β² = 28, find the value of k.

Accepted Answer

Step 1: By Vieta's formulas, α + β = 6 and αβ = k. Step 2: We know α² + β² = (α + β)² - 2αβ = 36 - 2k. Step 3: Given α² + β² = 28, so 36 - 2k = 28. Step 4: 2k = 8, thus k = 4. Step 5: Verify: x² - 6x + 4 = 0 has roots α = 3 + √5 and β = 3 - √5. Then α² + β² = (3+√5)² + (3-√5)² = 14 + 6√5 + 14 - 6√5 = 28 ✓

Question 22

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

Accepted Answer

Step 1: From x² - 5x + 6 = 0, we have α + β = 5 and αβ = 6. Step 2: We know (α - β)² = (α + β)² - 4αβ. Step 3: (α - β)² = 5² - 4(6) = 25 - 24 = 1.

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