Question 1

Simplify: (x + 4)(x - 4) using the difference of squares formula.

Accepted Answer

Step 1: Recognize the form (a + b)(a - b) = a² - b². Step 2: Here, a = x and b = 4. Step 3: (x + 4)(x - 4) = x² - 4² = x² - 16.

Question 2

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 3

If 4(x - 3) = 20, what is the value of x?

Accepted Answer

Step 1: Start with 4(x - 3) = 20. Step 2: Divide both sides by 4: x - 3 = 5. Step 3: Add 3 to both sides: x = 8. Therefore, x = 8.

Question 4

If (x + 3) is a factor of the polynomial p(x) = x² + 5x + 6, then p(-3) should equal:

Accepted Answer

Step 1: By the Factor Theorem, if (x + 3) is a factor, then p(-3) = 0. Step 2: Verify: p(-3) = (-3)² + 5(-3) + 6 = 9 - 15 + 6 = 0. Step 3: Therefore, p(-3) = 0.

Question 5

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 6

Which of the following is a root of the equation x² - 9 = 0?

Accepted Answer

Step 1: Factorize x² - 9 = 0 as (x - 3)(x + 3) = 0. Step 2: This gives x = 3 or x = -3. Step 3: Both 3 and -3 are roots. Step 4: Check the options for either value. Therefore, the answer is 3 (or -3 if listed).

Question 7

If 3x + 7 = 22, find the value of x.

Accepted Answer

Step 1: Start with 3x + 7 = 22. Step 2: Subtract 7 from both sides: 3x = 22 - 7 = 15. Step 3: Divide both sides by 3: x = 15 ÷ 3 = 5. Therefore, x = 5.

Question 8

If x + 1/x = 5, find the value of x² + 1/x².

Accepted Answer

Step 1: Use the identity (x + 1/x)² = x² + 2 + 1/x². Step 2: Rearrange: x² + 1/x² = (x + 1/x)² - 2. Step 3: Substitute: x² + 1/x² = 5² - 2 = 25 - 2 = 23.

Question 9

If 6x - 5 = 19, find x.

Accepted Answer

Step 1: Start with 6x - 5 = 19. Step 2: Add 5 to both sides: 6x = 24. Step 3: Divide both sides by 6: x = 4. Therefore, x = 4.

Question 10

Expand (2x + 3y)² using the algebraic identity.

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Here, a = 2x and b = 3y. Step 3: Apply: (2x + 3y)² = (2x)² + 2(2x)(3y) + (3y)² = 4x² + 12xy + 9y².

Question 11

What is the degree of the polynomial 4x⁵ + 3x³ - 2x + 7?

Accepted Answer

Step 1: Identify the highest power of x in the polynomial. Step 2: The terms are 4x⁵, 3x³, -2x, and 7. Step 3: The highest power is 5. Step 4: Therefore, the degree is 5.

Question 12

Find the remainder when p(x) = x³ + 2x² - 5x + 3 is divided by (x - 1).

Accepted Answer

Step 1: By the Remainder Theorem, when p(x) is divided by (x - a), the remainder is p(a). Step 2: Here, a = 1, so remainder = p(1). Step 3: p(1) = (1)³ + 2(1)² - 5(1) + 3 = 1 + 2 - 5 + 3 = 1.

Question 13

Simplify: (p + q)² + (p - q)² using algebraic identities.

Accepted Answer

Step 1: Expand (p + q)² = p² + 2pq + q². Step 2: Expand (p - q)² = p² - 2pq + q². Step 3: Add them: (p + q)² + (p - q)² = p² + 2pq + q² + p² - 2pq + q² = 2p² + 2q² = 2(p² + q²).

Question 14

Solve: 2x + 3 = x + 8

Accepted Answer

Step 1: Start with 2x + 3 = x + 8. Step 2: Subtract x from both sides: x + 3 = 8. Step 3: Subtract 3 from both sides: x = 5. Therefore, x = 5.

Question 15

Simplify: (x + 5)² - (x - 5)² using algebraic identities.

Accepted Answer

Step 1: Use the identity (a + b)² - (a - b)² = 4ab. Step 2: Here, a = x and b = 5. Step 3: Apply the formula: (x + 5)² - (x - 5)² = 4 × x × 5 = 20x.

Question 16

If (a + b)² = 49 and ab = 12, find the value of a² + b².

Accepted Answer

Step 1: Expand (a + b)² using the identity (a + b)² = a² + 2ab + b². Step 2: Substitute the given values: 49 = a² + 2(12) + b². Step 3: Simplify: 49 = a² + 24 + b². Step 4: Rearrange: a² + b² = 49 - 24 = 25.

Question 17

Solve: 3(2x + 1) = 21

Accepted Answer

Step 1: Start with 3(2x + 1) = 21. Step 2: Divide both sides by 3: 2x + 1 = 7. Step 3: Subtract 1 from both sides: 2x = 6. Step 4: Divide by 2: x = 3. Therefore, x = 3.

Question 18

If a - b = 7 and ab = 18, find the value of a² + b².

Accepted Answer

Step 1: Use the identity (a - b)² = a² - 2ab + b². Step 2: Rearrange to get a² + b² = (a - b)² + 2ab. Step 3: Substitute values: a² + b² = 7² + 2(18) = 49 + 36 = 85.

Question 19

Solve for y: 5y - 12 = 2y + 9

Accepted Answer

Step 1: Start with 5y - 12 = 2y + 9. Step 2: Subtract 2y from both sides: 3y - 12 = 9. Step 3: Add 12 to both sides: 3y = 21. Step 4: Divide by 3: y = 7. Therefore, y = 7.

Question 20

If the roots of x² + px + 12 = 0 are 3 and 4, what is the value of p?

Accepted Answer

Step 1: For a quadratic x² + px + 12 = 0 with roots 3 and 4, use sum of roots = -p. Step 2: Sum of roots = 3 + 4 = 7. Step 3: -p = 7, so p = -7. Step 4: Verify: product of roots = 3 × 4 = 12 ✓ (matches constant term). Therefore, p = -7.

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