SSC Quadratic Equations Questions with Solutions and Tricks

Quadratic equations are an important part of SSC algebra and are used in CGL Tier 1 and Tier 2, CHSL and many other exams. Most questions ask you to find the roots, use the sum and product of the roots or find a missing value with the help of the discriminant. If you remember three formulas and practise factorisation, each question takes less than a minute. This practice paper gives you 24 questions with answers and short solutions.

What Is a Quadratic Equation? Basic Concept in Simple Words

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a is not zero. It always has two roots (values of x that satisfy the equation). The roots may be real and different, real and equal, or not real at all.

• Factorisation: split the middle term and write the equation as (x - p)(x - q) = 0.
• Formula method: x = (-b plus or minus the square root of D) / 2a, where D = b^2 - 4ac.
• Nature of roots depends on the discriminant D.

Formulas and Shortcut Tricks

• Sum of roots = -b / a and product of roots = c / a.
• Discriminant D = b^2 - 4ac. If D > 0 the roots are real and different, if D = 0 they are real and equal, and if D < 0 there are no real roots.
• Equation from roots p and q: x^2 - (p + q)x + pq = 0.
• Useful identity: p^2 + q^2 = (p + q)^2 - 2pq.
• Equal roots condition: b^2 = 4ac.
• Check by substitution: put the root back in the equation to confirm the answer.

Set 1: Finding the Roots (Q1–Q8)

Q1. Find the roots of x^2 - 5x + 6 = 0.
Answer: 2 and 3
Solution: Factors: (x - 2)(x - 3) = 0.

Q2. Find the roots of x^2 - 7x + 12 = 0.
Answer: 3 and 4
Solution: (x - 3)(x - 4) = 0.

Q3. Find the roots of x^2 + 5x + 6 = 0.
Answer: -2 and -3
Solution: (x + 2)(x + 3) = 0.

Q4. Find the roots of x^2 - 9 = 0.
Answer: 3 and -3
Solution: x^2 = 9, so x = plus or minus 3.

Q5. Find the roots of x^2 - 4x = 0.
Answer: 0 and 4
Solution: x(x - 4) = 0.

Q6. Find the roots of 2x^2 - 5x + 2 = 0.
Answer: 2 and 1/2
Solution: 2x^2 - 4x - x + 2 = 2x(x - 2) - 1(x - 2) = (2x - 1)(x - 2).

Q7. Find the roots of x^2 + 2x - 15 = 0.
Answer: 3 and -5
Solution: (x + 5)(x - 3) = 0.

Q8. Find the roots of x^2 - 10x + 25 = 0.
Answer: 5 (equal roots)
Solution: (x - 5)^2 = 0, so both roots are 5.

Set 2: Sum, Product and Discriminant (Q9–Q16)

Q9. Find the sum of the roots of x^2 - 8x + 15 = 0.
Answer: 8
Trick: Sum = -b/a = 8.

Q10. Find the product of the roots of 2x^2 + 3x - 5 = 0.
Answer: -5/2
Trick: Product = c/a = -5/2.

Q11. Find the sum and product of the roots of 3x^2 - 6x + 2 = 0.
Answer: Sum = 2 and product = 2/3
Solution: Sum = 6/3 = 2 and product = 2/3.

Q12. Find the discriminant of x^2 - 6x + 9 = 0 and the nature of the roots.
Answer: D = 0, roots are real and equal
Solution: D = 36 - 36 = 0.

Q13. What is the nature of the roots of x^2 + x + 1 = 0?
Answer: No real roots
Solution: D = 1 - 4 = -3, which is negative.

Q14. What is the nature of the roots of x^2 - 5x + 4 = 0?
Answer: Real and different (1 and 4)
Solution: D = 25 - 16 = 9, which is positive.

Q15. For what value of k does x^2 - 4x + k = 0 have equal roots?
Answer: k = 4
Trick: D = 16 - 4k = 0, so k = 4.

Q16. For what values of k does x^2 + kx + 9 = 0 have equal roots?
Answer: k = 6 or -6
Solution: D = k^2 - 36 = 0, so k = plus or minus 6.

Set 3: Applications and Word Problems (Q17–Q24)

Q17. One root of x^2 - 7x + k = 0 is 3. Find k and the other root.
Answer: k = 12, other root = 4
Solution: 9 - 21 + k = 0 gives k = 12. Sum of roots is 7, so the other root = 4.

Q18. If the roots of x^2 - 5x + 6 = 0 are p and q, find p^2 + q^2.
Answer: 13
Trick: p + q = 5 and pq = 6. p^2 + q^2 = 25 - 12 = 13.

Q19. Form the quadratic equation whose roots are 2 and 5.
Answer: x^2 - 7x + 10 = 0
Solution: Sum = 7 and product = 10.

Q20. Form the quadratic equation whose roots are 3 and -4.
Answer: x^2 + x - 12 = 0
Solution: Sum = -1 and product = -12, so x^2 - (-1)x + (-12) = 0.

Q21. The sum of two numbers is 10 and their product is 21. Find the numbers.
Answer: 3 and 7
Solution: They are the roots of x^2 - 10x + 21 = 0, which gives (x - 3)(x - 7) = 0.

Q22. The product of two consecutive positive integers is 56. Find them.
Answer: 7 and 8
Solution: x(x + 1) = 56 gives x^2 + x - 56 = 0, so x = 7.

Q23. The area of a rectangle is 48 sq cm and its length is 2 cm more than its breadth. Find the dimensions.
Answer: Breadth = 6 cm and length = 8 cm
Solution: b(b + 2) = 48 gives b^2 + 2b - 48 = 0, so (b + 8)(b - 6) = 0 and b = 6.

Q24. If x + 1/x = 2, find x.
Answer: x = 1
Solution: x^2 - 2x + 1 = 0, so (x - 1)^2 = 0.

Common Mistakes in Quadratic Equations

• Forgetting to move all terms to one side before factorising.
• Using the wrong sign for the sum of the roots (it is -b/a).
• Ignoring the negative root in word problems when it is allowed, or including it when length cannot be negative.
• Making an error in D = b^2 - 4ac, especially with negative values of b or c.
• Not checking the roots by substitution.
• Cancelling a common variable and losing a root (for example in x^2 = 4x).

Related Topics to Study Next

• SSC Algebra Questions
• SSC Simplification Questions
• Surds and Indices Questions
• Fractions and Decimals Questions

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a second-degree equation of the form ax^2 + bx + c = 0, where a is not zero. It has two roots.

What is the formula to solve a quadratic equation?

The quadratic formula is x = (-b plus or minus the square root of (b^2 - 4ac)) / 2a.

What is the discriminant?

The discriminant is D = b^2 - 4ac. It tells whether the roots are real and different, real and equal or not real.

What is the sum and product of roots?

For ax^2 + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

How do you find the nature of roots?

Calculate D. If D is positive, the roots are real and different. If D is zero, they are equal. If D is negative, there are no real roots.

How do you form a quadratic equation from its roots?

Use x^2 - (sum of roots)x + (product of roots) = 0.

What is the condition for equal roots?

The roots are equal when b^2 = 4ac, that is, when D = 0.

How many quadratic equation questions come in SSC exams?

Usually one or two questions come in CGL Tier 1 and Tier 2 under algebra. In some exams they appear in the form of word problems.

Which is faster: factorisation or the formula?

Factorisation is faster when the numbers are small. Use the formula when the roots are not whole numbers.

Can a quadratic equation have only one root?

It has two roots always. When D = 0, both roots are equal, so it looks like one root.

How do I solve word problems on quadratic equations?

Assume the unknown as x, form an equation from the given condition, simplify it into ax^2 + bx + c = 0 and solve. Reject values that do not make sense.

How should I prepare quadratic equations for SSC exams?

Learn the formulas, practise factorisation, solve word problems and practise previous year algebra questions.