Quadratic Equation: A Complete Guide – Formulas, Methods, and Examples

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form:

ax² + bx + c = 0, where a ≠ 0

Here:

• a, b, and c are constants
• x is the variable

Roots of a Quadratic Equation

The roots (also called solutions) of a quadratic equation are the values of x for which the equation becomes zero.

Quadratic Formula

The most popular way to find the roots of a quadratic equation is using the Quadratic Formula:

x = (-b ± √(b² – 4ac)) / 2a

Proof of Quadratic Formula

To derive the quadratic formula, we complete the square:

Start with: ax² + bx + c = 0

1. Divide both sides by a: x² + (b/a)x + c/a = 0
2. Move constant to the other side: x² + (b/a)x = -c/a
3. Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = (b/2a)² – c/a
4. Factor and simplify: (x + b/2a)² = (b² – 4ac) / 4a²
5. Take square root: x + b/2a = ±√(b² – 4ac)/2a
6. Solve for x: x = (-b ± √(b² – 4ac)) / 2a

Nature of Roots of the Quadratic Equation

The nature of the roots depends on the value of the discriminant (D):

D = b² – 4ac

• If D > 0: Roots are real and distinct
• If D = 0: Roots are real and equal
• If D < 0: Roots are complex (non-real)

Sum and Product of Roots of Quadratic Equation

If α and β are the roots of the equation ax² + bx + c = 0, then:

• Sum (α + β) = -b/a
• Product (α × β) = c/a

Writing Quadratic Equations Using Roots

If the roots of the equation are α and β, then the quadratic equation can be written as:

x² – (α + β)x + αβ = 0

Formulas Related to Quadratic Equations

• Discriminant: D = b² – 4ac
• Roots: x = (-b ± √D) / 2a
• Sum of Roots: -b/a
• Product of Roots: c/a

Methods to Solve Quadratic Equations

1. Factorization Method
2. Completing the Square
3. Quadratic Formula
4. Graphical Method

Solving Quadratic Equations by Factorization

Example: x² + 5x + 6 = 0

Factor as: (x + 2)(x + 3) = 0 → x = -2 or -3

Method of Completing the Square

Example: x² + 6x + 5 = 0

Step 1: Move constant: x² + 6x = -5 Step 2: Add (6/2)² = 9 both sides: x² + 6x + 9 = 4 Step 3: Take square root: (x + 3)² = 4 → x = -3 ± 2 → x = -1 or -5

Graphing a Quadratic Equation

The graph of a quadratic equation y = ax² + bx + c is a parabola:

• Opens upward if a > 0
• Opens downward if a < 0

The points where the parabola cuts the x-axis are the roots.

Quadratic Equations Having Common Roots

If two quadratic equations have a common root, you can:

• Solve one equation to find the root
• Substitute into the second equation to verify

Maximum and Minimum Value of Quadratic Expression

For a quadratic expression y = ax² + bx + c:

• If a > 0: Minimum at x = -b/2a
• If a < 0: Maximum at x = -b/2a

Value: y = a(-b/2a)² + b(-b/2a) + c

Quadratic Equations Examples

1. Solve: x² – 7x + 10 = 0 → (x – 2)(x – 5) = 0 → x = 2, 5
2. Solve: 2x² + 4x + 2 = 0 → D = 0 → one root x = -1
3. Solve: x² + 2x + 5 = 0 → D < 0 → complex roots

20 FAQs on Quadratic Equation

1. What is a quadratic equation? An equation of the form ax² + bx + c = 0 where a ≠ 0.

2. What are the roots of a quadratic equation? Values of x that satisfy the equation.

3. What is the quadratic formula? (-b ± √(b² – 4ac)) / 2a

4. What is a discriminant? The expression b² – 4ac which determines the nature of roots.

5. What happens if D < 0? The equation has complex (non-real) roots.

6. Can a quadratic equation have one root? Yes, if D = 0.

7. Can roots be irrational? Yes, if D is not a perfect square.

8. What is meant by completing the square? A method to solve by turning the quadratic into a perfect square.

9. What is the vertex of a parabola? The point of max/min value: x = -b/2a

10. What is meant by common roots? Same root exists in two equations.

11. Why is a ≠ 0 in quadratic equation? If a = 0, the equation becomes linear.

12. Are quadratic equations used in real life? Yes, in physics, engineering, finance, etc.

13. What is the standard form? ax² + bx + c = 0

14. Is the graph of quadratic always a parabola? Yes.

15. How to check if an equation is quadratic? Look for degree 2 in variable.

16. What is meant by irrational roots? Roots that cannot be expressed as fractions.

17. Can roots be imaginary? Yes, if D < 0.

18. Is vertex same as root? No. Vertex is max/min point. Roots are x-intercepts.

19. What is the axis of symmetry? A vertical line passing through vertex: x = -b/2a

20. Are all quadratic equations solvable? Yes, though roots may be complex.

📌 Conclusion

Understanding quadratic equations is crucial for building strong algebra skills. From finding roots to analyzing the graph, mastering these concepts empowers students in academics and real-life problem-solving. Keep practicing and revisiting the formulas for success.

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