How to Solve Quadratic Equations Step by Step for Class 10

What Is a Quadratic Equation?

A quadratic equation is an algebraic equation of the second degree, meaning the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Solving a quadratic equation means finding the values of the variable (usually x) that satisfy the equation. These values are called the roots or solutions.

For Class 10 CBSE students, mastering quadratic equations is essential because they appear frequently in board exams and form the foundation for higher mathematics. There are three main methods to solve them: factorization, completing the square, and the quadratic formula.

Method 1: Solving by Factorization

Factorization is the most straightforward method when the quadratic expression can be split into two linear factors. This method works best when the roots are rational numbers.

Steps to Solve by Factorization

1. Write the equation in standard form: ax² + bx + c = 0
2. Find two numbers that multiply to give ac and add to give b
3. Split the middle term using these two numbers
4. Factor by grouping to get two binomial factors
5. Set each factor equal to zero and solve for x

Example: Solve x² + 5x + 6 = 0

Here, a = 1, b = 5, c = 6. We need two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.

x² + 5x + 6 = (x + 2)(x + 3) = 0

So x + 2 = 0 or x + 3 = 0, giving us x = −2 or x = −3.

Method 2: Completing the Square

Completing the square works for all quadratic equations. It involves rewriting the equation as a perfect square trinomial, which can then be solved by taking square roots.

Steps to Solve by Completing the Square

1. Write the equation in standard form: ax² + bx + c = 0
2. If a ≠ 1, divide the entire equation by a
3. Move the constant term to the right side
4. Add (b/2)² to both sides to complete the square on the left
5. Write the left side as a perfect square: (x + p)² = q
6. Take the square root of both sides and solve for x

Example: Solve x² + 6x + 5 = 0

Move the constant: x² + 6x = −5

Complete the square: x² + 6x + 9 = −5 + 9 = 4

(x + 3)² = 4

x + 3 = ±2

x = −3 + 2 = −1 or x = −3 − 2 = −5

Method 3: Using the Quadratic Formula

The quadratic formula is a universal method that solves any quadratic equation. It's derived from completing the square and is expressed as:

x = [−b ± √(b² − 4ac)] / 2a

The expression under the square root, b² − 4ac, is called the discriminant. It determines the nature of the roots.

What Is the Discriminant and Why Does It Matter?

The discriminant is b² − 4ac. If positive, the equation has two distinct real roots. If zero, there's one repeated real root. If negative, there are no real roots (only complex roots).

Steps to Solve Using the Quadratic Formula

1. Identify a, b, and c from the standard form ax² + bx + c = 0
2. Calculate the discriminant: b² − 4ac
3. Substitute a, b, and the discriminant into the formula
4. Simplify to find the two values of x

Example: Solve 2x² − 7x + 3 = 0

Here, a = 2, b = −7, c = 3.

Discriminant = (−7)² − 4(2)(3) = 49 − 24 = 25

x = [7 ± √25] / 4 = [7 ± 5] / 4

x = 12/4 = 3 or x = 2/4 = 0.5

Which Method Should You Use?

Use factorization when the quadratic factors neatly—it's quick and requires no complex calculations.

Use completing the square when you want to understand the geometric meaning of the equation or when the quadratic formula feels overwhelming.

Use the quadratic formula when factorization is difficult or when you need a reliable method that always works. Most Class 10 exams expect students to be comfortable with this method.

Common Mistakes to Avoid

Forgetting to set the equation to zero: Always rearrange to ax² + bx + c = 0 before solving.

Sign errors with the quadratic formula: Pay careful attention to the negative sign in front of b and the ± symbol.

Arithmetic mistakes with the discriminant: Double-check your calculation of b² − 4ac.

Incomplete solutions: Remember that a quadratic equation typically has two solutions (unless the discriminant is zero).

Practice Tips for Class 10 Students

Solving quadratic equations becomes easier with consistent practice. Start with equations that factor easily, then move to those requiring the quadratic formula. Work through important algebra topics for CBSE Class 10 to build a strong foundation.

If you're preparing for your board exams, consider exploring how to prepare for CBSE Class 10 maths board exam for a comprehensive study strategy. Practice at least 10–15 problems daily using each method. This repetition builds confidence and speed, both crucial during exams. Verify your answers by substituting them back into the original equation.

Real-World Applications

Quadratic equations model real situations like projectile motion, area optimization, and profit calculations in business. Understanding how to solve them opens doors to physics, engineering, and economics.

Summary

To solve quadratic equations for Class 10, write the equation in standard form ax² + bx + c = 0. Choose your method: factorization for simple cases, completing the square for deeper understanding, or the quadratic formula for reliability. Master all three methods and practice regularly to prepare for your board exams.