Question 1

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 and -3.

Question 2

Expand (2x + 3y)² and find the coefficient of xy.

Accepted Answer

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

Question 3

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 4

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 5

If 6x - 8 = 4x + 6, find x.

Accepted Answer

Step 1: Start with 6x - 8 = 4x + 6. Step 2: Subtract 4x from both sides: 2x - 8 = 6. Step 3: Add 8 to both sides: 2x = 14. Step 4: Divide by 2: x = 7. Therefore, the answer is 7.

Question 6

If α and β are roots of x² - 6x + 8 = 0, find the value of α² + β².

Accepted Answer

Step 1: From the equation, sum of roots α + β = 6 and product αβ = 8. Step 2: Use the identity α² + β² = (α + β)² - 2αβ. Step 3: α² + β² = (6)² - 2(8) = 36 - 16 = 20. Therefore, the answer is 20.

Question 7

Solve: 2x + 3x = 35

Accepted Answer

Step 1: Combine like terms on the left side: 5x = 35. Step 2: Divide both sides by 5: x = 7. Therefore, the answer is 7.

Question 8

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 9

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, the answer is 8.

Question 10

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

Accepted Answer

Step 1: Square both sides of x + 1/x = 5. Step 2: (x + 1/x)² = 25. Step 3: Expand using (a + b)² = a² + 2ab + b²: x² + 2(x)(1/x) + 1/x² = 25. Step 4: Simplify: x² + 2 + 1/x² = 25. Step 5: Therefore, x² + 1/x² = 25 - 2 = 23.

Question 11

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 12

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 13

If (a + b)² = 81 and ab = 18, 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: 81 = a² + 2(18) + b². Step 3: Simplify: 81 = a² + 36 + b². Step 4: Therefore, a² + b² = 81 - 36 = 45.

Question 14

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

Accepted Answer

Step 1: For a quadratic with roots α and β, the equation is x² - (α + β)x + αβ = 0. Step 2: Sum of roots = 3 + 4 = 7. Step 3: The equation becomes x² - 7x + 12 = 0. Step 4: Comparing with x² + px + 12 = 0, we get p = -7. Therefore, the answer is -7.

Question 15

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, the answer is 7.

Question 16

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 = 15. Step 3: Divide both sides by 3: x = 5. Therefore, the answer is 5.

Question 17

If a - b = 7 and ab = 12, find the value of (a - b)² + 4ab.

Accepted Answer

Step 1: We know (a - b)² = 49 (since a - b = 7). Step 2: We are given ab = 12. Step 3: Calculate (a - b)² + 4ab = 49 + 4(12) = 49 + 48 = 97.

Question 18

Solve for z: 3(z + 4) = 27

Accepted Answer

Step 1: Start with 3(z + 4) = 27. Step 2: Divide both sides by 3: z + 4 = 9. Step 3: Subtract 4 from both sides: z = 5. Therefore, the answer is 5.

Question 19

If a² + b² = 34 and ab = 15, find the value of (a + b)² - (a - b)².

Accepted Answer

Step 1: Expand (a + b)² - (a - b)² = (a² + 2ab + b²) - (a² - 2ab + b²) = 4ab. Step 2: Substitute ab = 15: (a + b)² - (a - b)² = 4 × 15 = 60.

Question 20

If x + 1/x = 5, find the value of x³ + 1/x³.

Accepted Answer

Step 1: Find x² + 1/x² using the identity (x + 1/x)² = x² + 2 + 1/x². So 25 = x² + 2 + 1/x², giving x² + 1/x² = 23. Step 2: Use the identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x). Step 3: x³ + 1/x³ = 5³ - 3(5) = 125 - 15 = 110.

SBI PO Algebra Questions

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