Quadratic Equation - Notes, Concept and All Important Formula

QUADRATIC EQUATION

1.  SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS :

(a) The solutions of the quadratic equation, $a x^{2}+b x+c=0$ is given by $\mathbf{x =\dfrac{- b \pm \sqrt{ b ^{2}-4 a c }}{ 2 a }}$

(b) The expression $b^{2}-4 a c \equiv D$ is called the discriminant of the quadratic equation.

(c) If $\alpha\, \& \, \beta$ are the roots of the quadratic equation $a x^{2}+b x+c=0$, then;

(i) $\alpha+\beta=-b / a$    (ii) $\alpha \beta= c / a$    (iii) $|\alpha-\beta|=\sqrt{ D } /| a |$

(d) Quadratic equation whose roots are $\alpha$  &  $\beta$ is $(x-\alpha)(x-\beta)=0$

i.e.  $x ^{2}-(\alpha+\beta) x +\alpha \beta=0$  i.e.  $x ^{2}-$ (sum of roots) $x +$ product of roots $=0$

(e) If $\alpha, \beta$ are roots of equation $a x^{2}+b x+c=0$, we have identity in $x$ as $a x^{2}+b x+c=a(x-\alpha)(x-\beta)$

---

All Chapter Notes, Concept and Important Formula

---

2.  NATURE OF ROOTS :

(a) Consider the quadratic equation $a x^{2}+b x+c=0$ where $a, b$, $c \in R \,\, \&\,\, a \neq 0$ then $;$

• $D >0 \Leftrightarrow$ roots are real & distinct (unequal).
• $D =0 \Leftrightarrow$ roots are real & coincident (equal)
• $D <0 \Leftrightarrow$ roots are imaginary.
• If $p + i q$ is one root of a quadratic equation, then the other root must be the conjugate $p - i q$ & vice versa. $(p, q \in R$  &  $i=\sqrt{-1})$

(b) Consider the quadratic equation $a x^{2}+b x+c=0$ where $a , b , c \in Q \,\, \&\,\,  a \neq 0$ then

• If $D$ is a perfect square, then roots are rational.
• If $\alpha= p +\sqrt{ q }$ is one root in this case, ( where $p$ is rational $\& \sqrt{ q }$ is a surd) then other root will be $p -\sqrt{ q }$.

---

---

3. COMMON ROOTS OF TWO QUADRATIC EQUATIONS

(a) Atleast one common root.

Let $\alpha$ be the common root of $a x^{2}+b x+c=0$  &  $a^{\prime} x^{2}+b^{\prime} x+c^{\prime}=0$

then $a \alpha^{2}+b \alpha+c=0 \& a^{\prime} \alpha^{2}+b^{\prime} \alpha+c^{\prime}=0 .$ By Cramer's

Rule $\dfrac{\alpha^{2}}{b c^{\prime}-b^{\prime} c}=\dfrac{\alpha}{a^{\prime} c-a c^{\prime}}=\dfrac{1}{a b^{\prime}-a^{\prime} b}$

Therefore, $\alpha=\dfrac{c a^{\prime}-c^{\prime} a}{a b^{\prime}-a^{\prime} b}=\dfrac{b c^{\prime}-b^{\prime} c}{a^{\prime} c-a c^{\prime}}$

So the condition for a common root is $\left(c a^{\prime}-c^{\prime} a\right)^{2}=\left(a b^{\prime}-a^{\prime} b\right)\left(b c^{\prime}-b^{\prime} c\right)$

(b) If both roots are same then $\dfrac{ a }{ a ^{\prime}}=\dfrac{ b }{ b ^{\prime}}=\dfrac{ c }{ c ^{\prime}}$

---

---

4. ROOTS UNDER PARTICULAR CASES

Let the quadratic equation $a x^{2}+b x+c=0$ has real roots and

(a) If $b =0 \Rightarrow$ roots are of equal magnitude but of opposite sign.

(b) If $c=0 \Rightarrow$ one roots is zero other is $-b / a$.

(c) If $a = c \Rightarrow$ roots are reciprocal to each other.

(d) If $a c<0 \Rightarrow$ roots are of opposite signs.

(e) If $\left.\begin{array}{r} a >0, b >0, c >0 \\ a <0, b <0, c <0\end{array}\right\} \Rightarrow$ both roots are negative.

(f) If $\left.\begin{array}{r}a>0, b<0, c>0 \\ a<0, b>0, c<0\end{array}\right\}  \Rightarrow$ both roots are positive.

(g) If sign of $a=$ sign of $b \neq$ sign of $c$ $\Rightarrow$ Greater root in magnitude is negative.

(h) If sign of $b=$ sign of $c \neq$ sign of a

$\Rightarrow$ Greater root in magnitude is positive.

(i) If $a+b+c=0 \Rightarrow$ one root is 1 and second root is $c / a$.

---

---

5. MAXIMUM & MINIMUM VALUES OF QUADRATIC EXPRESSION:

Maximum or Minimum Values of expression $y=a x^{2}+b x+c$ is $\dfrac{- D }{4 a }$ which occurs at $x =-( b / 2 a )$ according as $a <0$ or $a >0 .$

$y \in\left[\dfrac{-D}{4 a}, \infty\right)$ if $a>0 \quad \& \quad y \in\left(-\infty, \frac{-D}{4 a}\right]$ if $a<0 .$

---

---

6. LOCATION OF ROOTS :

Let $f ( x )= ax ^{2}+ bx + c$, where $a , b , c \in R , a \neq 0$

(a) Conditions for both the roots of $f ( x )=0$ to be greater than a specified number 'd' are $\mathbf{D \geq 0}$ and $\mathbf {a . f ( d )> 0 \,\, \& \,\, ( - b / 2 a )> d}$.

(b) Condition for the both roots of $f ( x )=0$ to lie on either side of the number 'd' in other words the number 'd' lies between the roots of $f(x)=0$ is $\mathbf{a . f ( d )< 0}.$

(c) Condition for exactly one root of $f ( x )=0$ to lie in the interval $( d , e)$ i.e. $d < x < e$ is $\mathbf{f ( d ) \cdot f ( e )< 0}$

(d) Conditions that both roots of $f(x)=0$ to be confined between the numbers d & e are (here $d < e ).$ 

$\mathbf{D \geq 0}$ and $\mathbf{a \cdot f ( d )> 0\,\,  \& \,\, a f ( e )> 0}$ and $\mathbf{d <(- b / 2 a )< e}$

---

---

7. GENERAL QUADRATIC EXPRESSION IN TWO VARIABLES :

$f(x, y)=a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c$ may be resolved into two linear factors if ;

$\triangle = abc + 2fgh – af^2 – bg^2– ch^2= 0$  OR   $\left|\begin{array}{lll}a & h & g \\h & b & f \\g & f & c\end{array}\right|=0$

---

---

8. THEORY OF EQUATIONS :

If $\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots \ldots \ldots \alpha_{n}$ are the roots of the equation;

$f(x)=a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots . .+a_{n-1} x+a_{n}=0$

where $a _{0}, a _{1}, \ldots \ldots a _{ n }$ are constants $a _{0} \neq 0$ then,

$\begin{aligned}\sum \alpha_{1} &=-\frac{ a _{1}}{ a _{0}}, \, \sum\alpha_{1} \alpha_{2}=+\frac{ a _{2}}{ a _{0}},\, \sum \alpha_{1} \alpha_{2} \alpha_{3} \\&=-\frac{ a _{3}}{ a _{0}}, \ldots \ldots, \alpha_{1}\alpha_{2} \alpha_{3} \ldots \ldots \alpha_{ n }(-1)^{ n } \frac{ a _{ n }}{ a _{0}}\end{aligned}$

Note:

(i) Every odd degree equation has at least one real root whose sign is opposite to that of its constant term, when coefficient of highest degree term is (+)ve {if not then make it (+)ve}.

$\text { Ex.} \,\, x^{3}-x^{2}+x-1=0$

(ii) Even degree polynomial whose constant term is (-)ve & coefficient of highest degree term is (+)ve has atleast two real roots, one (+)ve & one (-)ve.

(iii) If equation contains only even power of x & all coefficient are (+)ve, then all roots are imaginary.

(iv) Rational root theorem : If a rational number $\dfrac{ p }{ q }\left( p , q \in Z _{0}\right)$ is a root of polynomial equation with integral coefficient $a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots \ldots+a_{0}=0$, then $p$ divides $a_{0}$ and $q$ divides $a _{ n }$.      

---

---