Quadratic Expressions & Equations

by Suresh

QUADRATIC EXPRESSION: If $a \not=0, b, c$ are complex numbers then ax^2 + bx +c is called a quadratic expression in x.

QUADRATIC EQUATION: If $a \not=0,b,c$ are complex numbers then ax^2 + bx + c = 0 is called a quadratic equation in x.

ROOT OF A QUADRATIC EQUATION: If $a\alpha^2 + b\alpha + c = 0$ then $\alpha$ is a root or solution of the quadratic equation ax^2 + bx + c = 0.

A quadratic equation can not have more than two roots or two solutions. The roots of ax^2 + bx + c = 0 are $\frac{-b\pm \sqrt{b^2 - 4ac}}{2a}$ and its discrminent is $\triangle = b^2 -4ac$ .

NATURE OF THE ROOTS OF THE EQUATION

If a,b,c are real and $\triangle>0$ , then the roots are real and distinct.
If a,b,c are real and $\triangle=0$ , then the roots are real and equal.
If a,b,c are real and $\triangle<0$ , then the roots are two conjugate complex numbers.
If a,b,c are rational and $\triangle>0$ , and is a perfect square then the roots are rational and distinct.
If a,b,c are rational and $\triangle>0$ , and is not a perfect square then the roots are conjugate surds i.e $\alpha\pm \beta$ .
If a,b,c are rational and $\triangle<0$ , then the roots are conjugate complex numbers i.e, $\alpha\pm i\beta$ .

FORMATION OF THE QUADRATIC EQUATION WITH ROOTS $\alpha$ AND $\beta$ :
The quadratic equation whose roots are $\alpha$ and $\beta$ is
$x^2 - (\alpha + \beta)x + \alpha \beta = 0 \Rightarrow (x-\alpha)(x-\beta) = 0$ .

RELATION BETWEEN THE ROOTS $\alpha , \beta$ OF

1. $\alpha + \beta = \frac{-b}{a} , \alpha \beta = \frac{c}{a}.$

2. $|\alpha - \beta| = \frac {\sqrt{b ^2 - 4ac}}{|a|}.$

3. $\alpha ^2 + \beta ^2 = \frac{b ^2-2ac}{a ^2}.$

4. $\alpha ^3 + \beta ^3 = \frac{3abc-b ^3}{a ^3}.$

5. $\frac{1}{\alpha} + \frac{1}{\beta} = \frac{-b}{c}.$

6. $\frac{1}{\alpha ^2} + \frac{1}{\beta ^2} = \frac{b ^2 -2ac}{c ^2}.$

7. $\frac{1}{(a\alpha + b)} + \frac{1}{(a\beta+b)} = \frac{b}{ac}.$

8. $\frac{1}{(a\alpha + b) ^2} + \frac{1}{(a\beta +b) ^2} = \frac{ b ^2 -2ac}{a ^2 c ^2}.$

9. $\frac{1}{(a\alpha + b) ^3} + \frac{1}{(a\beta +b) ^3} = \frac{ b ^3 -3abc}{a ^3 c ^3}.$

10. $|\alpha ^2 - \beta ^2| = \frac{|b|\sqrt{b ^2 -4ac}}{a ^2}.$

PROPERTIES OF ROOTS OF THE EQUATION

If a and c are of the same sign i.e, $\frac{c}{a}$ is +ve, then both the roots are of same sign.

If a and c are of opposite sign i.e, $\frac{c}{a}$ is -ve, then the roots are of opposite sign.

If both the roots are -ve, then a,b,c will have the same sign.

If both the roots are +ve then a, c will have the same sign different from the sign of b.

If a=c, then the roots are reciprocal to each other.

If a+b+c=0,then the roots are 1 and $\frac{c}{a}.$

If a+c=b, then the roots are -1 and $\frac{-c}{a}.$

If the roots are in the ratio m:n then (m+n) ^2 ac = mnb ^2.

If one root is p times the other root then (1+p) ^2 ac = pb ^2.

If one root is equal to the n th power of the other root then
$(ac^n) ^\frac{1}{n+1} + (a ^nc) ^\frac{1}{n+1} +b = 0.$

If one root is square of the other, then a ^2 c + ac ^2 = b(3ac-b ^2).

If roots differ by unity, then b ^2 = 4ac + a ^2.

SAME ROOTS: If a_1 x ^2 + b_1 x + c_1 = 0 and a_2 x^2 + b_2 x + c_2 = 0 have the same roots then $\frac{a_1}{a_2} = \frac {b_1}{b_2} = \frac{c_1}{c_2}.$

ONE ROOT IS COMMON: The equations a_1 x ^2 + b_1 x + c_1 + 0 and a_2 x ^2 + b_2 x + c_2 = 0 where $a_1 b_2 - a_2 b_1 \not=0, a_1,a_2 \not=0$ , have one common root then (c_1 a_2 - c_2 a_1) ^2 = (a_1 b_2 - a_2 b_1)(b_1 c_2 - b_2 c_1) and the common root is
$\frac {c_1 a_2 - c_2 a_1}{a_1b_2 - a_2 b_1}.$

SIGNS OF AND

If the equation ax ^2 + bx + c = 0 has complex roots $(\triangle < 0)$ then and will have the same sign $\forall x \in R$

If the equation ax ^2 + bx + c = 0 has equal roots then and will have same sign $\forall x \in R - [\frac{-b}{2a}]$

If the equation ax ^2 + bx + c = 0 has real roots $\alpha , \beta (\triangle > 0, \alpha < \beta)$ then

1. $\alpha < x < \beta \Leftrightarrow a$ and ax ^2 + bx + c will have opposite sign.

2. $x<\alpha$ or $x > \beta \Leftrightarrow a$ and ax ^2 + bx + c will have same sign.

MAXIMUM OR MINIMUM VALUE O QUADRATIC EXPRESSION

If a > 0, then the minimum value of ax ^2 + bx + c is $\frac{4ac-b ^2}{4a}$ (This value is attained at $x = \frac{-b}{2a}$ ).

If a < 0, then the maximum value of ax ^2 + bx + c is $\frac{4ac-b ^2}{4a}$ (This value is attained at $x = \frac{-b}{2a}$ ).
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$If (\alpha < \beta), (x-\alpha)(x-\beta) \underline{<} 0$ $\Leftrightarrow x \in [\alpha, \beta] \quad i.e, \quad \alpha \underline{<} x \underline{<} \beta.$

$If (\alpha < \beta), (x-\alpha)(x-\beta) \underline{>}0$ $\Leftrightarrow x \in [-\infty, \alpha] \cup [\beta, \infty] i.e, x \underline{ <} \alpha or x \underline{>} \beta.$

If $\alpha, \beta$ are the roots of f(x)=ax ^2 + bx + c = 0 then the equation whose roots are

1. $-\alpha, -\beta \quad is \quad f(-x) = 0.$

2. $\frac{1}{\alpha}, \frac{1}{\beta} \quad is \quad f(\frac{1}{x}) = 0.$

3. $k\alpha, k\beta \quad(k \not=0) \quad is \quad f(\frac{x}{k}) = 0.$

4. $\alpha + k, \beta + k \quad is \quad f(x-k) = 0.$

5. $\alpha ^2, \beta ^2 \quad is \quad f(\sqrt{x}) = 0.$

6. $\alpha ^k, \beta ^k \quad is \quad f(\sqrt[k]{x}) = 0.$

7. $\frac{\alpha}{1+\alpha}, \frac{\beta}{1+\beta} \quad is \quad f(\frac{x}{1-x}) = 0.$

LOCATING THE ROOTS OF QUADRATIC EQUATION UNDER GIVEN CONDITIONS

Both the roots of equation ax^2 + bx + c = 0 are greater than a given number 'k' if

$\quad \triangle \underline > 0.$
$\quad af(k) > 0.$
$\quad\frac{-b}{2a} > k.$

Both the roots of equation ax^2 + bx + c = 0 are smaller than a given number 'k' if

$\quad \triangle \underline > 0.$
$\quad af(k) > 0.$
$\quad\frac{-b}{2a} < k.$

Exactly one root of ax^2 + bx + c = 0 lies between the numbers $'p' \quad and \quad 'q'$ if

$f(p)f(q)\underline<0$

but f(p) and f(q) are not simultaneously zero.

Both the roots of equation ax^2 + bx + c = 0 lie between two given numbers $'p'\quad and \quad 'q' \quad (p<q)$ if

$\quad \triangle \underline > 0.$
$\quad af(p) > 0.$
$\quad af(q) > 0.$
$\quad p < \frac{-b}{2a} < q.$

The extreme values of $f(x) = \frac{x ^2 - ax +b}{x ^2 + ax +b}$ are $f(\sqrt{-b}), f(\sqrt{b}).$
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