QUADRATIC EQUATION - Concept, Properties with Example

QUADRATIC EQUATION

INTRODUCTION

The algebraic expression of the form ax2 + bx + c, a 0 is called a quadratic expression, because the highest order term in it is of second degree. Quadratic equation means, ax2 + bx + c = 0. In general whenever one says zeroes of the expression ax2+bx + c, it implies roots of the equation ax2 + bx + c = 0, unless specified otherwise.

A quadratic equation has exactly two roots which may be real (equal or unequal) or imaginary.

SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS

The general form of quadratic equation is

${ax}^{2}+bx+c=0\, ,\, a\neq 0$

The roots can be found in following manner

$a({x}^{2}+\frac {b} {a}x+\frac {c} {a})=0$

$"{(x+\frac$

This expression can be directly used to find the two roots of a quadratic equation.

The expression b2 – 4 ac = D is called the discriminant of the quadratic equation
if $\alpha \, and\, \beta$ are the roots of the quadratic equation then

$(i).\, \alpha +\beta =\frac {-b} {a}$

$(ii).\, \alpha \beta =\frac {c} {a}$

$(iii)\, .\, \, \left | {\alpha -\beta } \right |={\sqrt {D}/\left | {a} \right |}$

Example :

Solution :

Nature of Roots :

(a). Consider the quadratic equation ax2 + bx + c = 0 where a,b,c are real number a!=0 then

$x=\frac {-b\pm \sqrt {D}} {2a}$

D > 0 => roots are real & distinct (unequal).
D = 0 => roots are real & coincident (equal).
D < 0 => roots are imaginary.
if p + i q is one root of a quadratic equation, then the other root must be the conjugate p–iq&vice versa.

(b).

Example :

Solution :

Example :

Find all the integral values of a for which the quadratic equation (x – a)(x – 10) + 1 = 0 has integral roots.

Solution :

Here the equation is

${x}^{2}-(a+10)x+10a+1\, =\, 0$

Since integral roots will always be rational it means D should be a perfect square.

$D={a}^{2}-20a+96$

If D is a perfect square it means we want difference of two perfect square as 4 which is possible only when

(a–10)(a-10) = 4 and D = 0.

(a-10) = +2 or (a-10) = -2

value of a should be 12, 8

ROOTS UNDER PARTICULAR CASES