Quadratic Equations - Revision Notes

 CBSE Class–10 Mathematics

Revision Notes

CHAPTER 04

QUADRATIC EQUATIONS

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1. Quadratic Equations

2. Solution by Factorisation

3. Solution by Completing the Square

4. Nature of Roots

1. The equation ax2+bx+c$a{x}^2+bx+c$ , a≠0$a\ne 0$  is the standard form of a quadratic equation, where a, b and c are real numbers.

ax2+bx+c=0,a≠0$a{x}^2+bx+c=0,a\ne 0$ 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 α$\alpha$  is said to be a root of the quadratic equation ax2+bx+c=0,a≠0$a{x}^2+bx+c=0,a\ne 0$ , a≠0$a\ne 0$. If aα2+bα+c=0,$a{\alpha }^2+b\alpha +c=0,$ the zeroes of quadratic polynomial ax2 + bx + c $\text{a}{\text{x}}^2+\text{ bx }+\text{ c }$ and the roots of the the quadratic equation ax2 + bx + c = 0$\text{a}{\text{x}}^2+\text{ bx }+\text{ c }=0$ are the same.

3. If we can factorise $$ax2 + bx + c = 0, a ≠ 0$\text{a}{\text{x}}^2+\text{ bx }+\text{ c }=0,\text{ a }\ne 0$ 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 ax2 + bx + c = 0$\text{a}{\text{x}}^2+\text{ bx }+\text{ c }=0$, a≠0$a\ne 0$ are given by −b±b2−4ac√2a,$\frac{-b\pm \sqrt{b^2-4ac}}{2a},$provided that b2− 4ac⩾ 0${\text{b}}^2-\text{ 4ac}⩾0$. It is called Quadratic formula.

5. A quadratic equation ax2 + bx + c = 0$\text{a}{\text{x}}^2+\text{ bx }+\text{ c }=0$, a≠0$a\ne 0$ has :

(a) Two distinct and real roots, if b2−4ac>0.$b^2-4ac>0.$

(b) Two equal and real roots, if b2−4ac=0.$b^2-4ac=0.$

(c) Two roots are not real, if b2−4ac<0.$b^2-4ac<0.$

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

(i) a2 + 2ab+ b2 = (a +b)2${\text{a}}^2+\text{ 2ab}+{\text{b}}^2={\left(\text{a }+\text{b}\right)}^2$

(ii) a2 - 2ab+ b2 = (a - b)2${\text{a}}^2\text{ - 2ab}+{\text{b}}^2={\left(\text{a - b}\right)}^2$

7. Discriminant of the quadratic equation ax2 + bx + c = 0$\text{a}{\text{x}}^2+\text{ bx }+\text{ c }=0$, a≠0$a\ne 0$ is givenby D=b2−4ac$D=b^2-4ac$.