Question 1

Simplify: (x + 4)(x - 4) - (x + 2)²

Accepted Answer

Step 1: Use (x + 4)(x - 4) = x² - 16 (difference of squares). Step 2: Use (x + 2)² = x² + 4x + 4. Step 3: Subtract: (x² - 16) - (x² + 4x + 4) = x² - 16 - x² - 4x - 4 = -4x - 20.

Question 2

Solve for y: 3(y - 4) = 15

Accepted Answer

Step 1: Expand the left side: 3y - 12 = 15. Step 2: Add 12 to both sides: 3y = 15 + 12 = 27. Step 3: Divide both sides by 3: y = 27 ÷ 3 = 9. Therefore, y = 9.

Question 3

Simplify: (a + b)² - (a - b)²

Accepted Answer

Step 1: Expand (a + b)² = a² + 2ab + b². Step 2: Expand (a - b)² = a² - 2ab + b². Step 3: Subtract: (a² + 2ab + b²) - (a² - 2ab + b²) = a² + 2ab + b² - a² + 2ab - b² = 4ab.

Question 4

If p - q = 5 and pq = 24, find the value of p² + q².

Accepted Answer

Step 1: Use the identity (p - q)² = p² - 2pq + q². Step 2: Rearrange: p² + q² = (p - q)² + 2pq. Step 3: Substitute: p² + q² = (5)² + 2(24) = 25 + 48 = 73.

Question 5

Expand: (2m + 3n)²

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Step 2: Here a = 2m and b = 3n. Step 3: (2m)² + 2(2m)(3n) + (3n)² = 4m² + 12mn + 9n².

Question 6

If a + b + c = 15 and ab + bc + ca = 72, find the value of a² + b² + c².

Accepted Answer

Step 1: Use the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca). Step 2: Rearrange: a² + b² + c² = (a + b + c)² - 2(ab + bc + ca). Step 3: Substitute: a² + b² + c² = (15)² - 2(72) = 225 - 144 = 81.

Question 7

If p(x) = x³ - 6x² + 11x - 6 and (x - 1) is a factor, find the other two factors.

Accepted Answer

Step 1: Since (x - 1) is a factor, divide p(x) by (x - 1) using polynomial division or factorization. Step 2: p(x) = (x - 1)(x² - 5x + 6). Step 3: Factor x² - 5x + 6 = (x - 2)(x - 3). Step 4: Therefore, the other two factors are (x - 2) and (x - 3).

Question 8

If x + y = 12 and xy = 35, find the value of x² + y².

Accepted Answer

Step 1: Use the identity (x + y)² = x² + 2xy + y². Step 2: Substitute: (12)² = x² + 2(35) + y². Step 3: 144 = x² + 70 + y². Step 4: x² + y² = 144 - 70 = 74.

Question 9

If 5x + 12 = 37, find the value of x.

Accepted Answer

Step 1: Start with 5x + 12 = 37. Step 2: Subtract 12 from both sides: 5x = 37 - 12 = 25. Step 3: Divide both sides by 5: x = 25 ÷ 5 = 5. Therefore, x = 5.

Question 10

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 term with the highest power is 4x⁵, which has power 5. Step 3: Therefore, the degree is 5.

Question 11

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 12

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 13

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 14

Find the roots of the equation 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. Therefore, the roots are 2 and 4.

Question 15

If the discriminant of the equation 3x² + bx + 4 = 0 is 0, find the value of b.

Accepted Answer

Step 1: For a quadratic ax² + bx + c = 0, discriminant D = b² - 4ac. Step 2: Here a = 3, c = 4, and D = 0. Step 3: b² - 4(3)(4) = 0. Step 4: b² - 48 = 0, so b² = 48. Step 5: b = ±√48 = ±4√3. Therefore, b = ±4√3.

Question 16

Solve for z: 6z + 8 = 2z + 24

Accepted Answer

Step 1: Subtract 2z from both sides: 6z - 2z + 8 = 24, which gives 4z + 8 = 24. Step 2: Subtract 8 from both sides: 4z = 24 - 8 = 16. Step 3: Divide both sides by 4: z = 16 ÷ 4 = 4. Therefore, z = 4.

Question 17

If (x + 3) is a factor of the polynomial p(x) = x² + 5x + 6, what is the value of p(-3)?

Accepted Answer

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

Question 18

What is the remainder when p(x) = 2x³ + 3x² - 5x + 4 is divided by (x - 1)?

Accepted Answer

Step 1: Use 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) = 2(1)³ + 3(1)² - 5(1) + 4 = 2 + 3 - 5 + 4 = 4.

Question 19

Simplify: (x + 2)(x - 2) - (x + 1)²

Accepted Answer

Step 1: Expand (x + 2)(x - 2) using difference of squares: x² - 4. Step 2: Expand (x + 1)² = x² + 2x + 1. Step 3: Subtract: (x² - 4) - (x² + 2x + 1) = x² - 4 - x² - 2x - 1 = -2x - 5.

Question 20

If 2x - 5 = x + 3, what is the value of x?

Accepted Answer

Step 1: Start with 2x - 5 = x + 3. Step 2: Subtract x from both sides: 2x - x - 5 = 3, which gives x - 5 = 3. Step 3: Add 5 to both sides: x = 3 + 5 = 8. Therefore, x = 8.

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