Quadratic Equation Complete Notes-Class 10,11 and 12

Quadratic Equation Complete Notes

Formula • Roots • Discriminant • α β Relations • Solved Examples

1. Introduction

Quadratic equation is one of the most important topics in algebra. It is used in mathematics, physics, engineering, economics and competitive examinations.

A quadratic equation is an equation whose highest power of variable is 2.

2. Definition of Quadratic Equation

The standard form of a quadratic equation is:

$$ax^2+bx+c=0$$

• \(a,b,c\) are real numbers
• \(a \neq 0\)
• \(x\) is variable
• \(a\) is coefficient of \(x^2\)
• \(b\) is coefficient of \(x\)
• \(c\) is constant term

3. Examples of Quadratic Equation

$$x^2-5x+6=0$$

$$2x^2+7x-3=0$$

$$3x^2-9=0$$

$$5x^2+4x+1=0$$

4. Roots of Quadratic Equation

The values of \(x\) satisfying the equation are called roots or zeroes.

If \( \alpha \) and \( \beta \) are roots:

$$a(x-\alpha)(x-\beta)=0$$

Every quadratic equation has two roots.

5. Quadratic Formula

Roots can be found using:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

This formula is applicable for every quadratic equation.

6. Discriminant

The quantity:

$$D=b^2-4ac$$

is called discriminant.

Discriminant determines the nature of roots.

Condition	Nature of Roots
\(D>0\)	Real and distinct roots
\(D=0\)	Real and equal roots
\(D<0 td="">	Imaginary roots

• If \(D\) is perfect square → rational roots
• If \(D\) is not perfect square → irrational roots

7. Relation Between Roots and Coefficients

For:

$$ax^2+bx+c=0$$

Sum of Roots

$$\alpha+\beta=-\frac{b}{a}$$

Product of Roots

$$\alpha\beta=\frac{c}{a}$$

8. Important α β Properties

Square of Roots

$$\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta$$

Cube of Roots

$$\alpha^3+\beta^3=(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$$

Difference of Roots

$$\alpha-\beta=\pm\frac{\sqrt{D}}{a}$$

Reciprocal Sum

$$\frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha\beta}$$

9. Formation of Quadratic Equation

If roots are \( \alpha \) and \( \beta \):

$$x^2-(\alpha+\beta)x+\alpha\beta=0$$

10. Condition for Equal Roots

$$b^2=4ac$$

11. Maximum and Minimum Value

$$y=ax^2+bx+c$$

• If \(a>0\) → minimum value exists
• If \(a<0 exists="" li="" maximum="" value="">

Value of x

$$x=-\frac{b}{2a}$$

Maximum / Minimum Value

$$\frac{-D}{4a}$$

12. Solved Examples

Example 1

Solve:

$$x^2-5x+6=0$$

$$(x-2)(x-3)=0$$

$$x=2,3$$

Example 2

Find roots using formula:

$$2x^2-7x+3=0$$

$$a=2,\quad b=-7,\quad c=3$$

$$x=\frac{-(-7)\pm\sqrt{49-24}}{4}$$

$$x=\frac{7\pm5}{4}$$

$$x=3,\frac12$$

Example 3

Find nature of roots:

$$x^2-4x+4=0$$

$$D=b^2-4ac$$

$$D=16-16=0$$

Roots are real and equal.

Example 4

Form equation whose roots are 2 and 5.

$$\alpha+\beta=7$$

$$\alpha\beta=10$$

$$x^2-7x+10=0$$

Example 5

Find minimum value:

$$y=x^2-6x+5$$

$$x=-\frac{-6}{2}=3$$

$$y=3^2-6(3)+5$$

$$y=-4$$

Minimum value = \(-4\)

Example 6

Find \(k\) if roots are equal:

$$x^2+6x+k=0$$

$$D=b^2-4ac$$

$$36-4k=0$$

$$k=9$$

13. Methods of Solving Quadratic Equation

• Factorization Method
• Completing Square Method
• Quadratic Formula Method

14. Applications of Quadratic Equation

• Projectile motion
• Area problems
• Business mathematics
• Engineering calculations
• Physics equations

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