A quadratic equation appears frequently in mathematics because it models countless real-world situations, from projectile motion to profit calculations. The formula for calculating a quadratic equation helps you find its roots quickly and accurately. No matter how complex the numbers look, this formula always works and gives exact solutions. Students preparing for CUET, SSC, Banking, JEE, or school exams use this method because it guarantees correct results.
Formula for Calculating Quadratic Equation Overview
| Formula | Variables | When It Is Used |
|---|---|---|
| x = (-b ± √(b² – 4ac)) / 2a | a, b, c are coefficients in ax² + bx + c = 0 | Used to find the roots (solutions) of quadratic equations |
What is a Quadratic Equation in Maths?
In mathematics, a quadratic equation is an equation of the form ax² + bx + c = 0, where a ≠ 0. The values that satisfy this equation are called roots. These roots may be real, equal, or imaginary depending on the value of the discriminant (D = b² – 4ac).
The quadratic formula is a universal tool for solving any quadratic equation.
To solve it:
- Write the equation in standard form: ax² + bx + c = 0.
- Identify the coefficients a, b, and c.
- Compute the discriminant: D = b² – 4ac.
- If D > 0 → two distinct real roots.
- If D = 0 → one repeated real root.
- If D < 0 → two complex roots.
- Apply the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
This formula is important in exams like CUET, SSC, Banking, and JEE. It also has real-life applications in engineering, physics, finance, and optimization problems.
Examples to Calculate Quadratic Equation
Example 1: Solve 2x² + 3x – 2 = 0
Step 1: Identify coefficients → a = 2, b = 3, c = –2
Step 2: Discriminant → D = b² – 4ac
= 3² – 4(2)(–2)
= 9 + 16
= 25
Step 3: Apply formula →
x = (-3 ± √25) / 4
= (-3 ± 5) / 4
- Case 1: (–3 + 5) / 4 = 2/4 = 0.5
- Case 2: (–3 – 5) / 4 = –8/4 = –2
Final Answer: x = 0.5 or x = –2
Example 2: Solve x² – 6x + 9 = 0
Step 1: Identify coefficients → a = 1, b = –6, c = 9
Step 2: Discriminant →
D = (–6)² – 4(1)(9)
= 36 – 36
= 0
Step 3: Apply formula →
x = (–b ± √D) / 2a
= (6 ± 0) / 2
= 6 / 2
= 3
Final Answer: x = 3 (repeated root)
FAQs about Quadratic Equation Formula
Q1. What is the quadratic formula?
It is x = (-b ± √(b² – 4ac)) / 2a, used to find the roots of any quadratic equation.
Q2. What is the discriminant?
It is b² – 4ac, and it determines whether the roots are real, equal, or imaginary.
Q3. Can quadratic equations have no real roots?
Yes, if b² – 4ac < 0, the roots are imaginary.
Q4. Why is the quadratic formula important in exams?
It works for every quadratic equation, making it essential for CUET, SSC, Banking, and JEE.
Q5. What is the degree of a quadratic equation?
The degree is 2, since x² is the highest power.
Q6. Is factoring easier than the formula?
Factoring is faster but only works sometimes. The quadratic formula always works.