Quadratic Equations Notes for SSC Exams

Quadratic equations are an important part of SSC Quantitative Aptitude, especially in Tier 2 and in bank-style questions. A quadratic equation can be solved in under a minute by factorisation, and the sum and product of roots save time in many questions. These quadratic equations notes for SSC explain every method with solved examples.

Standard Form

A quadratic equation has the form ax squared + bx + c = 0, where a is not zero. It has two roots (solutions). The roots can be real and different, real and equal, or not real.

Methods to Solve

• Factorisation: split the middle term bx into two terms whose product is ac and whose sum is b.
• Quadratic formula: x = (-b plus or minus the square root of (b squared - 4ac)) / 2a.
• Completing the square: rewrite the equation as (x + p) squared = q.

Important Formulas

Splitting the Middle Term

To factorise x squared - 5x + 6, find two numbers whose product is 6 and whose sum is -5. They are -2 and -3. So x squared - 2x - 3x + 6 = x(x - 2) - 3(x - 2) = (x - 2)(x - 3). The roots are 2 and 3. For a leading coefficient other than 1, multiply a and c first, then split the middle term, as in 2x squared + 7x + 3, where 2 x 3 = 6 and the numbers 6 and 1 add up to 7.

Special Cases and "I and II" Questions

In "I and II" questions, two quadratic equations are given and you have to compare the values of x and y. Solve both equations, write the roots on a number line and then compare. If every root of the first is greater than every root of the second, then x is greater than y. If the roots overlap, no relation can be established. Also remember that when c is zero, one root is always 0, and when b is zero, the roots are equal in size and opposite in sign.

Solved Examples

Example 1: Solve x squared - 5x + 6 = 0.
Solution: (x - 2)(x - 3) = 0, so x = 2 or 3.

Example 2: Solve 2x squared + 7x + 3 = 0.
Solution: Split 7x as 6x + x: 2x squared + 6x + x + 3 = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). So x = -1/2 or -3.

Example 3: Find the sum and product of the roots of x squared - 7x + 10 = 0.
Solution: Sum = -(-7)/1 = 7. Product = 10/1 = 10.

Example 4: Find the nature of the roots of x squared + 4x + 4 = 0.
Solution: D = 16 - 16 = 0, so the roots are real and equal (x = -2).

Example 5: Form the quadratic equation whose roots are 3 and -2.
Solution: Sum = 1, product = -6. The equation is x squared - x - 6 = 0.

Example 6: Solve x squared - 9 = 0.
Solution: x squared = 9, so x = 3 or -3.

Quick Tips to Solve Faster

• Try factorisation first because it is the fastest.
• Use the sum and product of the roots to answer questions without solving.
• Check the discriminant before looking for real roots.
• In "I and II" type questions, solve both equations and compare the roots on the number line.
• Check your roots by putting them back into the equation.

Common Mistakes to Avoid

• Using the wrong sign when splitting the middle term.
• Forgetting that a quadratic has two roots.
• Dividing both sides by x and losing a root.
• Mixing the formulas for the sum and product of roots.
• Using D less than 0 and still writing real roots.

Quick Revision Points

• Standard form: ax squared + bx + c = 0.
• Sum of roots = -b/a and product = c/a.
• D = b squared - 4ac.
• Equation from roots: x squared - (sum)x + product = 0.
• Factorise by splitting the middle term.

Practice Questions With Answers

1. Solve x squared - 7x + 12 = 0.
   Answer: x = 3 or 4 ((x - 3)(x - 4)).
2. Find the roots of x squared + x - 12 = 0.
   Answer: 3 and -4 ((x - 3)(x + 4)).
3. Find the sum of the roots of 2x squared - 10x + 8 = 0.
   Answer: 5 (10 / 2).
4. Find the nature of the roots of x squared - 4x + 5 = 0.
   Answer: No real roots (D = 16 - 20 = -4).
5. Find the roots of x squared - x - 20 = 0.
   Answer: 5 and -4 ((x - 5)(x + 4)).
6. If 2 is a root of x squared - 5x + k = 0, find k.
   Answer: 6 (4 - 10 + k = 0).
7. Find the product of the roots of 3x squared - 12x + 9 = 0.
   Answer: 3 (9 / 3).
8. Form the equation with roots 4 and 5. Find the coefficient of x.
   Answer: -9 (Sum = 9).
9. How many real roots does x squared + 2x + 1 = 0 have?
   Answer: One (repeated) (D = 0).
10. Find the positive root of x squared - 25 = 0.
    Answer: 5 (x squared = 25).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC algebra practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Algebra Formulas Notes
• Surds and Indices Notes
• Square Root and Cube Root Notes
• Simplification and BODMAS Notes
• SSC Algebra Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is an equation of the form ax squared + bx + c = 0, where a is not zero.

How many roots does a quadratic equation have?

It has two roots, which may be real and different, real and equal, or not real.

What is the quadratic formula?

x = (-b plus or minus the square root of (b squared - 4ac)) / 2a.

What is the discriminant?

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

What is the sum of the roots?

The sum of the roots is -b/a and the product is c/a.

How do I factorise a quadratic equation?

Split the middle term into two terms whose product is ac and whose sum is b, then take the common factors.

How many questions come from quadratic equations in SSC exams?

One or two questions come in Tier 2 and some in Tier 1, apart from the questions in bank exams.

How should I practise quadratic equations?

Practise factorisation daily and solve mixed sets of equations using the sum and product of roots.