Question 1

If 7x - 3 = 4x + 9, find x.

Accepted Answer

Step 1: Start with 7x - 3 = 4x + 9. Step 2: Subtract 4x from both sides: 7x - 4x - 3 = 9, which gives 3x - 3 = 9. Step 3: Add 3 to both sides: 3x = 9 + 3 = 12. Step 4: Divide by 3: x = 12 ÷ 3 = 4. Therefore, x = 4.

Question 2

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 3

Solve for y: 3(y - 4) = 18.

Accepted Answer

Step 1: Expand the left side: 3y - 12 = 18. Step 2: Add 12 to both sides: 3y = 18 + 12 = 30. Step 3: Divide both sides by 3: y = 30 ÷ 3 = 10. Therefore, y = 10.

Question 4

Solve: 4(x + 2) = 2(x + 8). What is x?

Accepted Answer

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

Question 5

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

Accepted Answer

Step 1: Factor the polynomial x² + 5x + 6. Step 2: Find two numbers that multiply to 6 and add to 5: these are 2 and 3. Step 3: x² + 5x + 6 = (x + 2)(x + 3). Step 4: Since (x + 3) is one factor, the other factor is (x + 2).

Question 6

Simplify: (2x + 3)(2x - 3)

Accepted Answer

Step 1: Recognize this as the difference of squares pattern (a + b)(a - b) = a² - b². Step 2: Here a = 2x and b = 3. Step 3: (2x)² - 3² = 4x² - 9.

Question 7

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 8

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 exponent is 5, so the degree is 5.

Question 9

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 10

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.

Question 11

Expand: (2x + 3y)(2x - 3y)

Accepted Answer

Step 1: Use the difference of squares identity (a + b)(a - b) = a² - b². Step 2: Here a = 2x and b = 3y. Step 3: (2x + 3y)(2x - 3y) = (2x)² - (3y)² = 4x² - 9y².

Question 12

Solve: (x + 5)/2 = 9. What is x?

Accepted Answer

Step 1: Start with (x + 5)/2 = 9. Step 2: Multiply both sides by 2: x + 5 = 9 × 2 = 18. Step 3: Subtract 5 from both sides: x = 18 - 5 = 13. Therefore, x = 13.

Question 13

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 values: (12)² = x² + 2(35) + y². Step 3: 144 = x² + y² + 70. Step 4: x² + y² = 144 - 70 = 74.

Question 14

If p(x) = x³ - 4x² + 5x - 2, find p(1).

Accepted Answer

Step 1: Substitute x = 1 into p(x) = x³ - 4x² + 5x - 2. Step 2: p(1) = (1)³ - 4(1)² + 5(1) - 2 = 1 - 4 + 5 - 2 = 0.

Question 15

What is the coefficient of x in the expansion of (x + 2)³?

Accepted Answer

Step 1: Expand (x + 2)³ using the binomial theorem or direct multiplication. Step 2: (x + 2)³ = x³ + 3(x²)(2) + 3(x)(2²) + 2³ = x³ + 6x² + 12x + 8. Step 3: The coefficient of x is 12.

Question 16

If m + n = 15 and m² + n² = 117, find the value of mn.

Accepted Answer

Step 1: Use the identity (m + n)² = m² + 2mn + n². Step 2: Substitute known values: (15)² = 117 + 2mn. Step 3: 225 = 117 + 2mn. Step 4: 2mn = 225 - 117 = 108. Step 5: mn = 54.

Question 17

If p - q = 8 and pq = 20, 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 values: p² + q² = (8)² + 2(20) = 64 + 40 = 104.

Question 18

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 19

Simplify: (x + 4)² - (x + 2)²

Accepted Answer

Step 1: Use the difference of squares identity a² - b² = (a + b)(a - b). Step 2: Let a = (x + 4) and b = (x + 2). Step 3: [(x + 4) + (x + 2)][(x + 4) - (x + 2)] = (2x + 6)(2) = 4x + 12.

Question 20

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

Accepted Answer

Step 1: Factorize x² - 5x + 6 = 0. Step 2: Find two numbers that multiply to 6 and add to -5: these are -2 and -3. Step 3: (x - 2)(x - 3) = 0. Step 4: x = 2 or x = 3. Therefore, the roots are 2 and 3.

SBI Clerk Algebra Questions

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