📐 Quadratic Equations — Practice Worksheet

Q1. The roots of the equation x² − 5x + 6 = 0 are:

1. 2 and 3
2. −2 and −3
3. 1 and 6
4. −1 and −6

Q2. The discriminant of the equation 2x² + 3x − 5 = 0 is:

1. 49
2. −31
3. 9
4. 41

Q3. If the sum of roots of x² − kx + 12 = 0 is 7, then k equals:

1. 5
2. 7
3. 12
4. −7

Q4. The nature of roots of x² + 4x + 5 = 0 is:

1. Real and equal
2. Real and distinct
3. Complex (imaginary)
4. Rational

Q5. If α and β are roots of x² − 3x + 2 = 0, then α² + β² equals:

1. 5
2. 9
3. 4
4. 7

Q6. The quadratic equation whose roots are 3 and −2 is:

1. x² − x − 6 = 0
2. x² + x − 6 = 0
3. x² − x + 6 = 0
4. x² + x + 6 = 0

Q7. For the equation x² − 2x + k = 0 to have equal roots, k must be:

1. 0
2. 1
3. 2
4. 4

Q8. If one root of x² − 6x + q = 0 is twice the other, then q equals:

1. 4
2. 8
3. 9
4. 12

Q9. The maximum value of the expression −x² + 4x − 3 is:

1. 1
2. 3
3. 4
4. 7

Q10. The number of real roots of x² + |x| + 1 = 0 is:

1. 0
2. 1
3. 2
4. 4

Q11. If α, β are roots of x² − px + q = 0, then (1/α + 1/β) equals:

1. p/q
2. q/p
3. −p/q
4. pq

Q12. The equation x² − 4x + 4 = 0 has:

1. Two distinct real roots
2. Two equal real roots, each equal to 2
3. No real roots
4. Two equal real roots, each equal to −2

Q13. Solve: 2x² − 7x + 3 = 0 using the quadratic formula.

Q14. If α and β are roots of 3x² − 5x + 2 = 0, find the value of α³ + β³.

Q15. Find the range of values of k for which the equation x² + 2(k − 1)x + (k + 5) = 0 has real roots.

Q16. The product of two consecutive positive integers is 132. Find the integers by forming a quadratic equation.

Q17. If the roots of x² + bx + c = 0 are in the ratio 2 : 3, express c in terms of b.

Q18. Find the quadratic equation whose roots are (2 + √3) and (2 − √3).

Q19. Solve the equation: √(2x + 1) = x − 1. Verify your solution(s).

Q20. For what values of k does the equation kx² − 6x + 2 = 0 have (a) equal roots, (b) no real roots?