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10 Mathematics notes Chapter 4 Quadratic Equations

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CBSE Class–10 Mathematics
Revision Notes
CHAPTER 04
QUADRATIC EQUATIONS

1. Quadratic Equations

2. Solution by Factorisation

3. Solution by Completing the Square

4. Nature of Roots

1. The equation $a{x^2} + bx + c$ , $a \ne 0$ is the standard form of a quadratic equation, where a, b and c are real numbers.

$a{x^2} + bx + c = 0,a \ne 0$ is known as Standard form or General form of a quadratic equation.

In other words, we can say that an equation of order (degree) 2 is called a quadratic equation.

2. A real number $\alpha$ is said to be a root of the quadratic equation $a{x^2} + bx + c = 0,a \ne 0$ , ${ a } \ne 0$ . If $a{\alpha ^2} + b\alpha + c = 0,$ the zeroes of quadratic polynomial ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c }}$ and the roots of the the quadratic equation ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c }} = {\text{ }}0$ are the same.

3. If we can factorise ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c }} = {\text{ }}0,{\text{ a }} \ne {\text{ }}0$ into product of two linear factors,then the roots of the quadratic equation can be found by equating each factors to zero.

4. The roots of a quadratic equation ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c }} = {\text{ }}0$ , ${ a } \ne 0$ are given by $\frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}},$ provided that ${{\text{b}}^2}-{\text{ 4ac}} \geqslant {\text{ }}0$ . It is called Quadratic formula.

5. A quadratic equation ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c }} = {\text{ }}0$ , ${ a } \ne 0$ has :

(a) Two distinct and real roots, if ${b^2} - 4ac\; > \;0.$

(b) Two equal and real roots, if ${b^2} - 4ac\; = \;0.$

6. A quadratic equation can also be solved by the method of completing the square.

(i) ${{\text{a}}^2}{\text{ }} + {\text{ 2ab}} + {\text{ }}{{\text{b}}^2}{\text{ }} = {\text{ }}{\left( {{\text{a }} + {\text{b}}} \right)^2}$

(ii) ${{\text{a}}^2}{\text{ - 2ab}} + {\text{ }}{{\text{b}}^2}{\text{ }} = {\text{ }}{\left( {{\text{a - b}}} \right)^2}$

7. Discriminant of the quadratic equation ${\text{a}}{{\text{x}}^2}{\text{ }} + {\text{ bx }} + {\text{ c }} = {\text{ }}0$ , ${ a } \ne 0$ is given by $D = {b^2} - 4ac$ .

Quadratic Equations class 10 Notes

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