Quadratic equations for Class 10 form a fundamental part of the CBSE Mathematics curriculum and assess a student’s understanding of algebraic concepts. A quadratic equation involves a second-degree polynomial and helps students find unknown values, identify the nature of solutions, and solve exam-oriented problems accurately. 

A clear understanding of how formulas work and how to apply them in different situations allows students to solve quadratic equations correctly and avoid common errors. In this blog, you will get to know the key quadratic equations formulas, how to use them to find roots, and how these formulas apply across different types of Class 10 problems.

Important Quadratic Equations Formulas Class 10

Standard Form of a Quadratic Equation

A quadratic equation represents a second-degree polynomial written as:

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are real numbers. The value of a controls the degree of the equation and confirms it as quadratic.

Roots of a Quadratic Equation

A real number α becomes a root of the equation ax² + bx + c = 0 when it satisfies the condition:

aα² + bα + c = 0

If this equality holds true, then x = α solves the quadratic equation.

Methods Used to Solve Quadratic Equations

Factorisation Method

This method works when the quadratic expression splits into two linear factors. After factorisation, equate each factor to zero to find the roots.

Best used when coefficients remain small and factor pairs exist clearly.

Completing the Square Method

This approach converts a quadratic expression into a perfect square by rearranging terms.

Key identities used:

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²

Quadratic Formula

When the discriminant value stays zero or positive, the roots follow this formula:

x = (−b ± √(b² − 4ac)) / 2a

Discriminant of a Quadratic Equation

The discriminant determines the type of roots and is defined as:

D = b² − 4ac

It provides immediate insight into the nature before solving the equation fully.

Nature of Roots Based on Discriminant

For the equation ax² + bx + c = 0:

• D > 0 → Two real and distinct roots
• D = 0 → Two equal real roots
• D < 0 → No real roots

Quadratic Equation from Given Roots

If the sum and product of roots are known, form the equation as:

x² − (sum of roots)x + (product of roots) = 0

Sum and Product of Roots

For the equation ax² + bx + c = 0:

• Sum of roots = −b / a
• Product of roots = c / a

Applications of Quadratic Equations Class 10

Quadratic equations model situations where values follow a curved or changing pattern.

• Area and Geometry: Students use quadratic equations to find dimensions of fields, rooms, and plots when area remains fixed.
• Motion and Speed: Quadratic equations help calculate time, distance, and speed in uniformly accelerated motion and braking problems.
• Business Mathematics: These equations support profit and loss calculations where output and cost vary.
• Engineering and Safety: Quadratic relationships appear in curve design and speed estimation during vehicle braking.

Tips to Learn Quadratic Equations Class 10

• Understand how a, b, and c affect the equation.
• Find the discriminant first to predict the roots.
• Choose factorisation for simple cases and formulas for complex ones.
• Write steps clearly to avoid calculation errors.
• Practice word problems and numerical questions regularly.

Example

Example 1: Find the discriminant of the quadratic equation
x² − 4x + 5 = 0 and state the nature of its roots.

Solution:
The given equation follows the form ax² + bx + c = 0, where
a = 1, b = −4, and c = 5.

The discriminant is
D = b² − 4ac
D = (−4)² − 4(1)(5)
D = 16 − 20
D = −4 < 0

Since the discriminant is less than 0, the quadratic equation has no real roots.