How to Solve a Quadratic Equation Step by Step (With Examples)

What Is a Quadratic Equation?

A quadratic equation is an equation where the highest power of the unknown is 2. Its standard form is ax^2 + bx + c = 0, where a, b and c are numbers and a is not zero. The letter x is the unknown we want to find.

Quadratic equations show up in Class 10 NCERT maths (the chapter on quadratic equations), in board exams and in JEE and other entrance tests. They also appear in real problems about area, speed, profit and falling objects.

This guide shows you how to solve a quadratic equation step by step using three methods: factorisation, the quadratic formula and completing the square. You can check your answers with a free quadratic equation calculator at any time.

Step Zero: Write the Equation in Standard Form

Before you choose any method, move every term to the left side so the right side is zero. Then arrange the terms in the order x^2, x, number.

For example, x^2 + 3x = 10 becomes x^2 + 3x – 10 = 0. Here a = 1, b = 3 and c = -10. Always note the signs, because a wrong sign is the most common reason for a wrong answer.

Method 1: Solve by Factorisation

1. Write the equation in standard form.
2. Find two numbers whose product is a x c and whose sum is b.
3. Use these numbers to split the middle term.
4. Group the terms and take out common factors.
5. Set each bracket equal to zero and solve for x.

This method is fast when the numbers are small and whole. It does not work well when the roots are fractions with ugly decimals, and then you use the formula.

Method 2: Solve by the Quadratic Formula

The formula works for every quadratic equation. For ax^2 + bx + c = 0:

x = (-b + sqrt(b^2 – 4ac)) / 2a and x = (-b – sqrt(b^2 – 4ac)) / 2a

The part under the square root, D = b^2 – 4ac, is called the discriminant. It tells you what kind of roots to expect before you finish the work.

Discriminant D	Type of roots	Example
D greater than 0	Two different real roots	x^2 – 5x + 6 = 0 (D = 1)
D equal to 0	Two equal real roots	x^2 – 6x + 9 = 0 (D = 0)
D less than 0	No real roots	x^2 + 4x + 7 = 0 (D = -12)

Method 3: Solve by Completing the Square

1. Divide the whole equation by a, so the coefficient of x^2 becomes 1.
2. Move the constant term to the right side.
3. Add the square of half the coefficient of x to both sides.
4. Write the left side as a perfect square, such as (x + p)^2.
5. Take the square root of both sides, remembering both the plus and minus values.
6. Solve for x.

This is the method from which the quadratic formula itself is derived, so it is worth understanding even if you rarely use it in exams.

How to Choose the Right Method

Students often waste time by picking a long method for an easy equation. This small table can guide you.

Situation	Best method	Reason
Small whole numbers, a = 1	Factorisation	Two numbers can be found by inspection.
Large or awkward numbers	Quadratic formula	It works every time without guessing.
Need to find the turning point or prove the formula	Completing the square	It shows the structure of the equation.
No b term, such as x^2 – 49 = 0	Direct square root	Move 49 across and take the root: x = 7 or -7.

A Handy Check: Sum and Product of Roots

For ax^2 + bx + c = 0, the sum of the two roots is -b/a, and the product of the roots is c/a. You can use this to test your answers quickly.

Take x^2 – 5x + 6 = 0, where the roots are 2 and 3. The sum is 2 + 3 = 5, and -b/a = 5. The product is 2 x 3 = 6, and c/a = 6. Both match, so the roots are right.

The same check works for the plot problem solved below: the roots 15 and -18 add up to -3, and -b/a = -3. They multiply to -270, which is c/a. This short test takes ten seconds and can save a whole mark in an exam.

Three Worked Examples with Indian Numbers

Example 1: Factorisation, x^2 – 5x + 6 = 0

Here a = 1, b = -5 and c = 6. We need two numbers with product 6 and sum -5. They are -2 and -3.

Split the middle term: x^2 – 2x – 3x + 6 = 0. Group: x(x – 2) – 3(x – 2) = 0. Take the common bracket: (x – 2)(x – 3) = 0. So x = 2 or x = 3.

Check x = 2: 4 – 10 + 6 = 0. Check x = 3: 9 – 15 + 6 = 0. Both work.

Example 2: Formula, the plot of land problem

A rectangular plot in a housing society has a length 3 m more than its width. Its area is 270 square metres. Find the width and length.

Let the width be w metres. Then the length is w + 3, and the area is w(w + 3) = 270. This gives w^2 + 3w – 270 = 0, so a = 1, b = 3, c = -270.

Discriminant: D = 3^2 – 4(1)(-270) = 9 + 1080 = 1089. The square root of 1089 is 33. Now w = (-3 + 33) / 2 = 15 or w = (-3 – 33) / 2 = -18.

A width cannot be negative, so we reject -18. The width is 15 m and the length is 18 m. Check: 15 x 18 = 270. Correct.

Example 3: Completing the square, x^2 + 6x – 7 = 0

Move the constant: x^2 + 6x = 7. Half of 6 is 3, and its square is 9. Add 9 to both sides: x^2 + 6x + 9 = 16.

The left side is a perfect square: (x + 3)^2 = 16. Take the square root: x + 3 = 4 or x + 3 = -4. So x = 1 or x = -7.

Check x = 1: 1 + 6 – 7 = 0. Check x = -7: 49 – 42 – 7 = 0. Both roots are correct.

What if the discriminant is negative?

Try x^2 + 4x + 7 = 0. Here D = 16 – 28 = -12. You cannot take the square root of a negative number among real numbers, so the equation has no real roots. In Class 10 you simply write “no real roots”.

Key takeaway: Choose the method by looking at the numbers. Try factorising for 30 seconds. If it does not work, go straight to the formula, because it never fails.

Benefits of Learning to Solve Quadratics

• Exam marks: Quadratic equations carry steady marks in CBSE Class 10 boards and in JEE algebra.
• Real problem solving: Area, speed, age and profit problems often turn into quadratics.
• Logical thinking: Choosing between three methods trains you to plan before you calculate.
• Base for higher maths: Parabolas, projectile motion and optimisation all rest on this topic.
• Self-checking: Substituting the roots back lets you prove your own answer is right.

Challenges and Limitations

Each method has weak spots. Factorisation fails when the roots are irrational or the numbers are large. Completing the square needs careful fractions when a is not 1. The formula is reliable, but one sign slip inside the square root ruins the answer.

Quadratic equations with negative discriminants have no real roots, which can confuse beginners. Also, word problems add the extra step of turning a story into an equation, and that step is often harder than the algebra. When roots are decimals, such as 3.732, rounding rules matter, and the decimal calculator can help you check the arithmetic.

Common Mistakes to Avoid

• Forgetting to set the right side to zero: You cannot factorise x^2 + 3x = 10 directly. Move 10 across first.
• Losing the sign of b or c: In x^2 – 5x + 6, b is -5, not 5. Write a, b and c on the side before substituting.
• Dropping the minus root: Taking a square root gives two answers, plus and minus.
• Squaring b wrongly: b^2 is always positive, so (-5)^2 = 25, not -25.
• Dividing by x: Cancelling x from both sides can lose the root x = 0. Factorise instead.
• Skipping the check: A quick substitution catches most slips and shows which root the story rejects.

Best Practices

• Write a, b and c clearly, with signs, before you start.
• Calculate the discriminant first. It tells you if work is even needed.
• Try factorisation first for small whole numbers, then switch to the formula.
• Keep the working neat, one step per line.
• Always substitute the roots back into the original equation.
• Use an online calculator after solving by hand to confirm, never instead of the working.

Conclusion

Solving a quadratic equation is a short routine: standard form, choose a method, find the roots, check. Factorisation is quick, the formula is dependable, and completing the square explains why the formula works.

Practise one problem of each type this week, including a word problem like the plot of land. Within a few days the steps will feel automatic.

Frequently Asked Questions

What is the easiest way to solve a quadratic equation?

Factorisation is the easiest when the numbers are small and the roots are whole numbers. If you cannot factorise quickly, use the quadratic formula, which always works. Completing the square is mainly useful for understanding where the formula comes from.

What does the discriminant tell us?

The discriminant is b^2 – 4ac. If it is positive, there are two different real roots. If it is zero, the two roots are equal, and if it is negative, there are no real roots.

Can a quadratic equation have only one solution?

Yes, when the discriminant is zero, the two roots are equal, so you get one repeated solution. For example, x^2 – 6x + 9 = 0 gives x = 3 only.

Why do we sometimes reject a root?

In word problems, answers like negative lengths or negative ages make no sense. So you keep only the root that fits the real situation, as we did with the plot width in the example above.

Is the quadratic equation taught in CBSE?

Yes. Quadratic equations are a chapter in NCERT Class 10 maths, and the ideas are used again in Class 11 and in JEE-level problems.