Quadratic equation

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04 QUADRATIC EQUATION

Quadratic equation: A quadratic equation is equation of the form

a x^{2} + b x + c = 0, a \neq 0, a, b, c a r e r e a l n u m b e r s .

• T h e p o l y n o m i a l a x^{2} + b x + c h a s d e g r e e 2 .

Quadratic equation can have at most two roots.

e . g . 2 x^{2} + 3 x + 4 = 0, x^{2} - 3 x + 4 = 0

S o l u t i o n o f a q u a d r a t i c e q u a t i o n a x^{2} + b x + c :

Factorisation method

• F a c t o r i s e t e r m a \times c

• S p l i t m i d d l e t e r m s u \sin g s u i t a b l e f a c t o r s o f a \times c

Get two linear factors

Equate each factor with zero,get the roots of

• E . g x^{2} + 5 x + 6 = 0

x^{2} - 5 x + 6

x^{2} - 3 x - 2 x + 6 = 0 . . . . . . . (f a c t o r s o f 6 = 3 \times 2)

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

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

x - 3 = 0, x - 2 = 0

x = 3, x = 2

Solution by completing square:

• W r i t e e q u a t i o n i n t h e f o r m o f a x^{2} + b x = - c

A d d o n b o t h s i d e ({\frac{b}{2 \sqrt{a}}}^{2}) : i . e . a x^{2} + b x + {(\frac{b}{2 \sqrt{a}})}^{2} = - c + {(\frac{b}{2 \sqrt{a}})}^{2}

• S i m p l i f y : i . e . {(\sqrt{a x} + \frac{b}{2 \sqrt{a}})}^{2} = \frac{- 4 a c + b^{2}}{4 a}

• T a k e s q u a r e r o o t o n b o t h s i d e : i . e . \sqrt{a} x + \frac{b}{2 \sqrt{a}} = \pm (\frac{\sqrt{b^{2} - 4 a c}}{2 a})

• S i m p l i f y a n d f i n d v a l u e o f x . i . e ., x = \frac{- b \pm (\sqrt{b^{2} - 4 a c})}{2 a}

e . g . 2 x^{2} - 5 x + 3 = 0

2 x^{2} - 5 x = - 3

2 x^{2} - 5 x + {(\frac{5}{2 \sqrt{2}})}^{2} = - 3 + {(\frac{5}{2 \sqrt{2}})}^{2}

2 x^{2} - 5 x + \frac{25}{8} = - 3 + \frac{25}{8}

{(\sqrt{2} x - \frac{5}{2 \sqrt{2}})}^{2} = \frac{25 - 24}{8}

(\sqrt{2} x - \frac{5}{2 \sqrt{2}}) = \pm \frac{1}{2 \sqrt{2}}

\sqrt{2} x = \frac{1}{2 \sqrt{2}} + \frac{5}{2 \sqrt{2}} o r \sqrt{2} x = - \frac{1}{2 \sqrt{2}} + \frac{5}{2 \sqrt{2}}

\sqrt{2} x = \frac{3}{\sqrt{2}} o r \sqrt{2} x = \frac{2}{\sqrt{2}}

x = \frac{3}{2} o r x = \frac{2}{2} = 1

x = \frac{3}{2}, 1

Solution by formula method:

The roots of quadratic equation

a x^{2} + b x + c = 0 a r e x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}

Nature of roots:

C a s e 1 : b^{2} - 4 a c = 0, r o o t s a r e r e a l a n d e q u a l .

C a s e 2 : b^{2} - 4 a c > 0, r o o t s a r e r e a l a n d u n e q u a l .

C a s e 3 : b^{2} - 4 a c < 0, n o r e a l r o o t s .

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