Quadratic Equations
Quadratic equations are equations in which the variable is raised to the power of 2. They appear naturally in problems involving area, motion, projectiles, business profit models, and optimization.
This lesson explains what quadratic equations are, their forms, methods of solving, graphical meaning, and real-world applications — step by step.
What Is a Quadratic Equation?
A quadratic equation is an equation of degree 2, meaning the highest power of the variable is 2.
It always involves a squared term like x².
Example: x² + 5x + 6 = 0
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is:
ax² + bx + c = 0
Here, a ≠ 0, and a, b, c are constants.
Meaning of the Coefficients
Each coefficient in a quadratic equation has a role. Understanding them helps in graphs and real-life interpretation.
- a → controls the shape of the curve
- b → controls the direction and position
- c → constant term (y-intercept)
Why Quadratic Equations Matter
Quadratic equations model situations where change is not constant but curved.
They appear when things accelerate, bend, or reach maximum/minimum values.
Graph of a Quadratic Equation (Parabola)
The graph of a quadratic equation is called a parabola.
It is a smooth U-shaped or inverted U-shaped curve.
Roots (Solutions) of a Quadratic Equation
The roots of a quadratic equation are the values of x that make the equation equal to zero.
Graphically, roots are where the parabola crosses the x-axis.
Method 1: Factorization
Factorization means expressing the quadratic equation as a product of two linear factors.
Example: x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = −2 or x = −3
When Factorization Works
Factorization works when the middle term can be split easily.
It is fast and preferred in exams when possible.
Method 2: Completing the Square
Completing the square rewrites the equation into a perfect square form.
This method is very useful for understanding graphs.
Example: x² + 6x + 5 = 0
x² + 6x = −5
Add (6/2)² = 9 to both sides
(x + 3)² = 4
x = −1 or −5
Method 3: Quadratic Formula
The quadratic formula works for all quadratic equations. It is the most powerful and universal method.
Formula:
x = [ −b ± √(b² − 4ac) ] / 2a
Understanding the Discriminant
The expression (b² − 4ac) is called the discriminant.
It tells us how many solutions the equation has.
| Discriminant | Nature of Roots |
| > 0 | Two distinct real roots |
| = 0 | One real repeated root |
| < 0 | No real roots |
Quadratic Equations in Real Life
Quadratic equations appear in situations involving curves and optimization.
- Projectile motion
- Maximum profit problems
- Area calculations
- Physics and engineering models
Quadratic Equations in Competitive Exams
Exams test:
- Identifying coefficients
- Choosing the right solving method
- Understanding roots and discriminant
Common Mistakes to Avoid
Most mistakes happen due to algebraic carelessness.
- Sign errors in formula
- Incorrect factorization
- Forgetting a ≠ 0
Practice Questions
Q1. Solve: x² − 7x + 10 = 0
Q2. Find the discriminant of: x² + 4x + 8 = 0
Q3. Solve using formula: x² + 2x − 3 = 0
Quick Quiz
Q1. What is the shape of a quadratic graph?
Q2. Which method works for all quadratics?
Quick Recap
- Quadratic equations have degree 2
- Standard form: ax² + bx + c = 0
- Three solving methods exist
- Graphs are parabolas
- Widely used in real life and exams