Quadratic equations

The general form of a quadratic equation over real numbers is $a{x}^2+bx+c=0$ where $a,b,c\in R\mathrm{\&}a\ne$ 0 . The solution of the quadrati equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ is given by

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. The expression $b^2-4ac$ is called the discriminant of the quadratic equation and is denoted by D.

Nature of roots : For the quadratic equation $a{x}^2+bx+c=0$, where $a,b,c\in R$ and $a\ne 0$, then

For the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ where $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{Q}$ and $\mathrm{a}\ne 0$, then :

Note 1: If $\alpha$ is a root of $f(\mathrm{x})=0$, then the polynomial $f(\mathrm{x})$ is exactly divisible by $\mathrm{x}-\alpha$ or $(\mathrm{x}-\alpha )$ is a factor of $f(\mathrm{x})$ and vice versa.

Note 2: $x^2+bx+c=0$ cannot have three different roots. If it has, then the equation becomes an identity in $\mathrm{x}$. ie, $\mathrm{a}=\mathrm{b}=\mathrm{c}=0$.

Relation between roots and coefficients

If ${\alpha }_1,{\alpha }_2,\dots \dots \dots {\alpha }_n$ are roots of the equation $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{n-2}{x}^{n-2}+\dots \dots \dots a_2{x}^2+a_{1}x+a_0=0$, then $f(\mathrm{x})={\mathrm{a}}_{\mathrm{n}}(\mathrm{x}-{\alpha }_1)(\mathrm{x}-{\alpha }_2)(\mathrm{x}-{\alpha }_3)$ $(\mathrm{x}-{\alpha }_{\mathrm{n}})$

$\therefore {\mathrm{a}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}+{\mathrm{a}}_{\mathrm{n}-1}{\mathrm{x}}^{\mathrm{n}-1}+{\mathrm{a}}_{\mathrm{n}-2}{\mathrm{x}}^{\mathrm{n}-2}+\dots \dots \dots \dots +{\mathrm{a}}_2{\mathrm{x}}^2+{\mathrm{a}}_1\mathrm{x}+{\mathrm{a}}_0={\mathrm{a}}_{\mathrm{n}}(\mathrm{x}-{\alpha }_1)(\mathrm{x}-{\alpha }_2)$.

Comparing the coefficients of like powers of $x$ an both sides, we get,

$\begin{array}{rl}& S_1={\alpha }_1+{\alpha }_2+\dots \dots \dots \dots +{\alpha }_n=\sum {\alpha }_i=\frac{-a_{n-1}}{a_n}=\frac{-\text{ coeft. of }{x}^{n-1}}{\text{ coeft of }{x}^n}\\ & S_2={\alpha }_1{\alpha }_2+{\alpha }_1{\alpha }_3+\dots \dots \dots \dots \dots \sum _{i\ne j}{\alpha }_i{\alpha }_j=(-1{)}^2\frac{a_{n-2}}{a_n}=(-1{)}^2\frac{\text{ coeft. of }{x}^{n-2}}{\text{ coeft of }{x}^n}\\ & S_3={\alpha }_1{\alpha }_2{\alpha }_3+{\alpha }_2{\alpha }_3{\alpha }_4+\dots \dots \dots \dots \dots \sum _{i\ne j\ne k}{\alpha }_i{\alpha }_j{\alpha }_k=(-1{)}^3\frac{a_{n-3}}{a_n}=(-1{)}^3\frac{\text{ coeft. of }{x}^{n-3}}{\text{ coeft of }{x}^n}\end{array}$

${\mathrm{S}}_{\mathrm{n}}={\alpha }_1{\alpha }_2{\alpha }_3\dots \dots \dots \dots \dots \dots \dots {\alpha }_n=(-1{)}^{\mathrm{n}}\frac{{\mathrm{a}}_0}{{\mathrm{a}}_{\mathrm{n}}}=(-1{)}^{\mathrm{n}}\frac{\text{ constant term }}{\mathrm{coeft}\text{ of }{\mathrm{x}}^{\mathrm{n}}}$

Here ${\mathrm{S}}_{\mathrm{k}}$ denotes the sum of the products of the roots taken ’ $\mathrm{k}$ ’ at a time.

Particular cases :-

Quadratic equation : If $\alpha \mathrm{\&}\beta$ are the roots of the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$, then ${\mathrm{S}}_1=\alpha +\beta =\frac{-\mathrm{b}}{\mathrm{a}},\mathrm{\&}{\mathrm{S}}_2=\alpha \beta =\frac{\mathrm{c}}{\mathrm{a}}$

Cubic equation: If $\alpha ,\beta ,\gamma$ are the roots of the cubic equation ${\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+\mathrm{d}=0$, then

$\begin{array}{rl}& {\mathrm{S}}_1=\alpha +\beta +\gamma =\frac{-b}{\mathrm{a}}\\ & {\mathrm{S}}_2=\alpha \beta +\beta \alpha +\gamma \alpha =(-1{)}^2\frac{\mathrm{c}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{a}}\\ & {\mathrm{S}}_3=\alpha \beta \gamma =(-1{)}^3\frac{\mathrm{d}}{\mathrm{a}}=\frac{-\mathrm{d}}{\mathrm{a}}\end{array}$

Biquadratic equation : If $\alpha ,\beta ,\gamma ,\delta$ are roots of the biquadratic equation ${\mathrm{ax}}^4+{\mathrm{bx}}^3+{\mathrm{cx}}^2+\mathrm{dx}+\mathrm{e}=0$, then

$\begin{array}{rl}& {\mathrm{S}}_1=\alpha +\beta +\gamma +\delta =\frac{-b}{\mathrm{a}}\frac{\text{ constant term }}{{\text{ coeft of }}^{\mathrm{n}}}\\ & {\mathrm{S}}_2=\alpha \beta +\beta \gamma +\alpha \delta +\beta \gamma +\beta \delta +\gamma \delta =(-1{)}^2\frac{\mathrm{c}}{\mathrm{a}}\\ & \text{ Or }{\mathrm{S}}_2=(\alpha +\beta )(\gamma +\delta )+\alpha \beta +\gamma \delta =\frac{\mathrm{c}}{\mathrm{a}}\\ & {\mathrm{S}}_3=\alpha \beta \gamma +\beta \gamma \delta +\gamma \delta \alpha +\alpha \beta \delta =(-1{)}^3\frac{\mathrm{d}}{\mathrm{a}}\\ & \text{ Or }{\mathrm{S}}_3=\alpha \beta (\gamma +\delta )+\gamma \delta (\alpha +\beta )=\frac{-\mathrm{d}}{\mathrm{a}}\\ & \text{ and }{\mathrm{S}}_4=\alpha \beta \gamma \delta =(-1{)}^4\frac{\mathrm{e}}{\mathrm{a}}=\frac{\mathrm{e}}{\mathrm{a}}\end{array}$

Formation of a polynomial equation from given roots

If ${\alpha }_1,{\alpha }_2,{\alpha }_3,\dots \dots \dots \dots \dots ,{\alpha }_n$ are the roots of an ${\mathrm{n}}^{\text{th }}$ degree equation, then the equation is ${\mathrm{x}}^{\mathrm{n}}-{\mathrm{S}}_1{\mathrm{x}}^{\mathrm{n}-1}+{\mathrm{S}}_2{\mathrm{x}}^{\mathrm{n}-2}-{\mathrm{S}}_3{\mathrm{x}}^{\mathrm{n}-3}+\dots \dots \dots \dots .+(-1{)}^{\mathrm{n}}{\mathrm{S}}_{\mathrm{n}}=0$ where ${\mathrm{S}}_{\mathrm{k}}$ denotes the sum of the products of roots taken $\mathrm{k}$ at a time.

Particular cases

Quadratic equation : If $\alpha ,\beta$ are the roots of a quadratic equation, then the equation is $x^2-S_{1}x+S_2=0$ ie, $x^2-(\alpha +\beta )x+\alpha \beta =0$.

Cubic equation : If $\alpha ,\beta ,\gamma$ are the roots of a cubic equation. Then the equation is, ${\mathrm{x}}^3-{\mathrm{S}}_1{\mathrm{x}}^2+{\mathrm{S}}_2\mathrm{x}-{\mathrm{S}}_3=0$ ie, ${\mathrm{x}}^3-(\alpha +\beta +\gamma ){\mathrm{x}}^2+(\alpha \beta +\beta \gamma +\gamma \alpha )\mathrm{x}-\alpha \beta \gamma =0$

Biquadratic equation : If $\alpha ,\beta ,\gamma ,\delta$ are the roots of a biquadratic equation, then the equation is ${\mathrm{x}}^4-{\mathrm{S}}_1{\mathrm{x}}^3+{\mathrm{S}}_2{\mathrm{x}}^2-{\mathrm{S}}_3\mathrm{x}+{\mathrm{S}}_4=0$

ie, $x^4-(\alpha +\beta +\gamma +\delta )x^3+(\alpha \beta +\beta \gamma +\gamma \delta +\alpha \delta +\beta \delta +\alpha \gamma )x^2-$ $(\alpha \gamma \beta +\alpha \beta \delta +\beta \gamma \delta +\alpha \gamma \delta )x+\alpha \beta \gamma \delta =0$.

Quadratic Expression : An expression of the form $a{x}^2+bx+c$, where $a,b,c\in R\mathrm{\&}a\ne 0$ is called a quadratic expression in $\mathrm{x}$. So in general quadratic expression is represented as: $f(\mathrm{x})={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ or $\mathrm{y}=$ $a{x}^2+bx+c$.

Graph of a quadratic Expression

Let $\mathrm{y}={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ where $\mathrm{a}\ne 0$.

Then $y=a(x^2+\frac{b}{a}x+\frac{c}{a})\Rightarrow y=a(x^2+\frac{bx}{a}+\frac{b^2}{4a^2}+\frac{c}{a}-\frac{b^2}{4a^2})$

$\Rightarrow y+\frac{b^2-4ac}{4a}=a{(x+\frac{b}{2a})}^2\Rightarrow y+\frac{D}{4a}=a{(x+\frac{b}{2a})}^2$

Let $y+\frac{D}{4a}=Y\mathrm{\&}x+\frac{b}{2a}=X$

$\therefore \mathrm{Y}={\mathrm{aX}}^2$ or ${\mathrm{X}}^2=\frac{\mathrm{Y}}{\mathrm{a}}$

Clearly it is the equation of a parabola having its vertex at $(\frac{-b}{2a},\frac{-D}{4a})$.

If $\mathrm{a}>0$, then the parabola open upwards.

If $a<0$, then the parabola open downwards.

Sign of quadratic Expression

(1) The parabola will intersect the $x$-axis in two distinct points if $\mathrm{D}>0$.

(i) $\mathrm{a}>0$

Let $f(\mathrm{x})=0$ have 2 real roots

$\alpha \mathrm{\&}\beta (\alpha <\beta )$. Then $f(x)>0$

$\mathrm{\forall }\mathrm{x}\in (-\mathrm{\infty },\alpha )\cup (\beta \mathrm{\infty })$ and $f(\mathrm{x})<0$

$\mathrm{\forall }\mathrm{x}\in (\alpha ,\beta )$

(ii) $\mathrm{a}<0$

Let $f(\mathrm{x})=0$ have 2 real roots

$\alpha \mathrm{\&}\beta (\alpha <\beta )$ Then $f(\mathrm{x})<0$

$\mathrm{\forall }\mathrm{x}\in (-\mathrm{\infty },\alpha )\cup (\beta \mathrm{\infty })$

$\mathrm{\&}f(\mathrm{x})>0$ for all $\mathrm{x}\in (\alpha ,\beta )$

(2) The parabola will touch the $\mathrm{x}$-axis at one point if $\mathrm{D}=0$

(3) The parabola will not intersect $\mathrm{x}$-axis if $\mathrm{D}<0$.

NOTE : Condition that a quadratic function $f(\mathrm{x},\mathrm{y})={\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2f\mathrm{y}+\mathrm{c}$ may be resolved into two linear factions is that

$abc+2fgh-a{f}^2-b^2-c{h}^2=\left|\begin{array}{lll}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{h}& \mathrm{b}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right|=0$

NOTE :

(i) For $\mathrm{a}>0,f(\mathrm{x})={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ has least value at $\mathrm{x}=\frac{-\mathrm{b}}{2\mathrm{a}}$. This least value is given by $\frac{-\mathrm{D}}{4\mathrm{a}}$

(ii) For $\mathrm{a}<0,f(\mathrm{x})={\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}$ has greatest value at $\mathrm{x}=\frac{-\mathrm{b}}{2\mathrm{a}}$. This greatest value is given by $\frac{-\mathrm{D}}{4\mathrm{a}}$

Solved Examples

1. If $\alpha ,\beta$ are roots of ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0;\alpha +\mathrm{h}$ and $\beta +\mathrm{h}$ are roots of ${\mathrm{px}}^2+\mathrm{qx}+\mathrm{r}=0$ and ${\mathrm{D}}_1,{\mathrm{D}}_2$ are their discriminants, then ${\mathrm{D}}_1:{\mathrm{D}}_2=$

(a). $\frac{{\mathrm{a}}^2}{{\mathrm{p}}^2}$

(b). $\frac{b^2}{q^2}$

(c). $\frac{{\mathrm{c}}^2}{{\mathrm{r}}^2}$

(d). None of these

2. If $a\in Z$ and the equation $(x-a)(x-10)+1=0$ has integral roots, then the values of $a$ are

(a). 10, 8

(b). 12, 10

(c). 12, 8

(d). None of these

3. If $\alpha ,\beta$ are roots of the equation $(x-a)(x-b)+c=0(c\ne 0)$, then then roots of the equation ( $x-c-$ $\alpha )(\mathrm{x}-\mathrm{c}-\beta )=\mathrm{c}$ are

(a). $a$ and $b+c$

(b). $a+\mathrm{c}$ and $b$

(c). $a+c$ and $b+c$

(d). None of these

4. Let ${\mathrm{\Delta }}^2$ be the discriminant and $\alpha ,\beta$ be the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$. Then $2\mathrm{a}\alpha +\mathrm{\Delta }$ and $2\mathrm{a}\beta -\mathrm{\Delta }$ can be roots of the equation

(a). ${\mathrm{x}}^2+2\mathrm{bx}+{\mathrm{b}}^2=0$

(b). ${\mathrm{x}}^2-2\mathrm{bx}+{\mathrm{b}}^2=0$

(c). $x^2+2bx-3b^2-16ac=0$

(d). ${\mathrm{x}}^2+2\mathrm{bx}-3{\mathrm{b}}^2+16\mathrm{ac}=0$

Show Answer

Solution : $\alpha ,\beta =\frac{-b\pm \sqrt{{\mathrm{\Delta }}^2}}{2a}$

$\therefore$ quadatic equation is

• ${\mathrm{x}}^2+2\mathrm{bx}-3{\mathrm{b}}^2+16\mathrm{ac}=0$

$\therefore$ quadratic equation is

• ${\mathrm{x}}^2+2b\mathrm{bx}+{\mathrm{b}}^2=0$

Answer : (a and d)

5. The polynomial equation $(a{x}^2+bx+c)(a{x}^2-dx-c)=0,ac\ne 0$ has

(a). four real roots

(b). atleast two real roots

(c). atmost two real roots

(d). No real roots

6. If $\alpha ,\beta$ are roots of $x^2-p(x+1)-q=0$, then the value of $\frac{{\alpha }^2+2\alpha +1}{{\alpha }^2+2\alpha +q}+\frac{{\beta }^2+2\beta +1}{{\beta }^2+2\beta +q}$ is

(a). 1

(b). 2

(c). 3

(d). None of these

7. Let $\alpha ,\beta ,\gamma$ be the roots of the equation $x^3+4x+1=0$, then $(\alpha +\beta {)}^{-1}+(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}$ equals

(a). 2

(b). 3

(c). 4

(d). 5

Practice questions

1. The minimum value of $f(\mathrm{x})={\mathrm{x}}^2+2\mathrm{bx}+2{\mathrm{c}}^2$ is greatest than the maximum value of $g(x)=-x^2-2cx+b^2$, then ( $x$ being a real )

(a). $|c|>\frac{|b|}{\sqrt{3}}$

(b). $\frac{|\mathrm{c}|}{\sqrt{2}}>|\mathrm{b}|$

(c). $-1<\mathrm{c}<\sqrt{2}\mathrm{b}$

(d). Non real value of b & c exist

2. If $\mathrm{P}(\mathrm{x})$ is a polynomial of degree less than or equal to 2 and $\mathrm{S}$ is the set of all such polynomials so that $\mathrm{P}(1)=1,\mathrm{P}(0)=0$ and ${\mathrm{P}}^1(\mathrm{x})>0\mathrm{\forall }\mathrm{x}\in [0,1]$, then $\mathrm{S}=$ Here, $\mathrm{P}(\mathrm{x})=\mathrm{b}{\mathrm{x}}^2+\mathrm{ax}+\mathrm{c}$

(a). $\varphi$

(b). $\{(1-\mathrm{a}){\mathrm{x}}^2+\mathrm{ax},0<\mathrm{a}<2\}$

(c). $\{(1-\mathrm{a}){\mathrm{x}}^2+\mathrm{ax},\mathrm{a}>0\}$

(d). $\{(1-\mathrm{a}){\mathrm{x}}^2+\mathrm{ax},0<\mathrm{a}<1\}$

3. In the quadratic equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ if $\mathrm{\Delta }={\mathrm{b}}^2-4\mathrm{ac}$ and $\alpha +\beta ,{\alpha }^2+{\beta }^2$ and ${\alpha }^3+{\beta }^3$ are in G.P, where $\alpha ,\beta$ are the roots of the equation, then

(a). $\mathrm{\Delta }\ne 0$

(b). ${\mathrm{b}}_{\mathrm{\Delta }}=0$

(c). ${\mathrm{c}}_{\mathrm{\Delta }}=0$

(d). $\mathrm{\Delta }=0$

4. If $a,b,c$ are the sides of a triangle $ABC$ such that $x^2-2(a+b+c)x+3\lambda (ab+bc+ca)=0$ has real roots, then

(a). $\lambda <\frac{4}{3}$

(b). $\lambda >\frac{5}{3}$

(c). $\lambda \in (\frac{4}{3},\frac{5}{3})$

(d). $\lambda \in (\frac{1}{3},\frac{5}{3})$

5. Let $\alpha \mathrm{\&}\beta$ be the roots of $x^2-6x-2=0$, with $\alpha >\beta$. If $a_n={\alpha }^n-{\beta }^n$ for $n\ge 1$, then the value of $\frac{{\mathrm{a}}_{10}-2{\mathrm{a}}_8}{2{\mathrm{a}}_9}$ is

(a). 1

(b). 2

(c). 3

(d). 4

6. If $x^2-10ax-11b=0$ have roots $c\mathrm{\&}d\cdot x^2-10cx-11d=0$ have roots $a\mathrm{\&}b$, then $a+b+c+d$ is

(a). 1210

(b). 1120

(c). 1200

(d). None of these

7. If ${\mathrm{t}}_{\mathrm{n}}$ denotes the ${\mathrm{n}}^{\text{th }}$ term of an A.P. and ${\mathrm{t}}_{\mathrm{p}}=\frac{1}{\mathrm{q}}$ and ${\mathrm{t}}_{\mathrm{q}}=\frac{1}{\mathrm{p}}$, then which of the following is necessarily a root of the equation $(p+2q-3r)x^2+(q+2r-3p)x+(r+2p-3q)=0$ is

(a). $t_p$

(b). ${\mathrm{t}}_{\mathrm{q}}$

(c). $t_{\mathrm{pq}}$

(d). $t_{p+q}$

8. The curve $y=(\lambda +1)x^2+2$ intersect the curve $y=\lambda x+3$ in exactly one point, if $\lambda$ equals

(a). $\{-2,2\}$

(b). $\{1\}$

(c). $\{-2\}$

(d). $\{2\}$

9. Read the passage and answer the following questions.

Consider the equation $x^4+(1-2k)x^2+k^2-1=0$ where $k$ is real. If $x^2$ is imaginary, or $x^2<0$, the equation has no real roots. If $x^2>0$, the equation has real roots.

(i) The equation has no real roots if $\mathrm{k}\in$

(a). $(-\mathrm{\infty }-1)$

(b). $(-1,1)$

(c). $(1,\frac{5}{4})$

(d). $(\frac{5}{4},\mathrm{\infty })$

(ii) The equation has only two real roots if $\mathrm{k}\in$

(a). $(-\mathrm{\infty }-1)$

(b). $(0,1)$

(c). $(1,2)$

(d). $(-1,1)$

(iii) The equation has four real roots if $\mathrm{k}\in$

(a). $(-\mathrm{\infty },0)$

(b). $(-1,1)$

(c). $(1,\frac{5}{4})$

(d). $(1,\mathrm{\infty })$

10. If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$, then the value of $\left|\begin{array}{ccc}1& \cos (\beta -\alpha )& \cos \alpha \\ \cos (\alpha -\beta )& 1& \cos \beta \\ \cos \alpha & \cos \beta & 1\end{array}\right|$ is

(a). $\sin (\alpha +\beta )$

(b). $\sin \alpha \sin \beta$

(c). $1+\cos (\alpha +\beta )$

(d). None of these

11. If $(1+k){\tan }^{2}x-4\tan x-1+k=0$ has real roots, then

(a). ${\mathrm{k}}^2\le 5$

(b). $\tan ({\mathrm{x}}_1+{\mathrm{x}}_2)=2$

(c). for $\mathrm{k}=2,{\mathrm{x}}_1=\frac{\pi }{4}$

(d). for $\mathrm{k}=1,{\mathrm{x}}_1=0$

12. If $p,q\in \{1,2,3,4\}$, the number of equations of the form ${\mathrm{px}}^2+\mathrm{qx}+1=0$ having real roots is

(a). 15

(b). 9

(c). 7

(d). 8

13. In $\mathrm{△}\mathrm{PQR}\mathrm{\angle }\mathrm{R}=\frac{\pi }{2}$. If $\tan \frac{\mathrm{p}}{2}\mathrm{\&}\tan \frac{\mathrm{Q}}{2}$ are the roots of the equation ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0(\mathrm{a}\ne 0)$ then

(a). $\mathrm{a}+\mathrm{b}=\mathrm{c}$

(b). $\mathrm{b}+\mathrm{c}=0$

(c). $a+\mathrm{c}=\mathrm{b}$

(d). $\mathrm{b}=\mathrm{c}$