-
Engineering and Architecture
Exams
Colleges
Predictors
Resources
-
Computer Application and IT
Quick Links
Colleges
-
Pharmacy
Colleges
Resources
-
Hospitality and Tourism
Colleges
Resources
Diploma Colleges
-
Competition
Other Exams
Resources
-
School
Exams
Top Schools
Products & Resources
-
Study Abroad
Top Countries
Resources
-
Arts, Commerce & Sciences
Colleges
Upcoming Events
Resources
-
Management and Business Administration
Exams
Colleges & Courses
Predictors
-
Learn
Law Preparation
MBA Preparation
Engineering Preparation
Medical Preparation
-
Online Courses and Certifications
Top Streams
Specializations
- Digital Marketing Certification Courses
- Cyber Security Certification Courses
- Artificial Intelligence Certification Courses
- Business Analytics Certification Courses
- Data Science Certification Courses
- Cloud Computing Certification Courses
- Machine Learning Certification Courses
- View All Certification Courses
Resources
-
Medicine and Allied Sciences
Colleges
Predictors
Resources
-
Law
Resources
Colleges
-
Animation and Design
Exams
Predictors & Articles
Colleges
Resources
-
Media, Mass Communication and Journalism
Colleges
Resources
-
Finance & Accounts
Top Courses & Careers
Colleges
Relation Between Roots and Coefficient of Quadratic Equation - (Concept)
Let alpha and beta be two roots of a quadratic equation. So, we have
$\begin{aligned} & \alpha=\frac{-b-\sqrt{D}}{2 a} \\ & \beta=\frac{-b+\sqrt{D}}{2 a}\end{aligned}$
The sum of roots:
$\alpha+\beta=\frac{-\mathrm{b}-\sqrt{\mathrm{D}}}{2\mathrm{a}}+\frac{\mathrm{b}+\sqrt{\mathrm{D}}}{2 \mathrm{a}}=\frac{-\mathrm{b}}{\mathrm{a}}$
Product of roots:
$\begin{aligned} & \alpha \cdot \beta=\left(\frac{-\mathrm{b}-\sqrt{\mathrm{D}}}{2 \mathrm{a}}\right) \cdot\left(\frac{-\mathrm{b}+\sqrt{\mathrm{D}}}{2\mathrm{a}}\right)\\&=\frac{\mathrm{b}^2-\mathrm{D}}{4\mathrm{a}^2}=\frac{\mathrm{b}^2-\mathrm{b}^2+4\mathrm{ac}}{4\mathrm{a}^2}=\frac{4 \mathrm{ac}}{4 \mathrm{a}^2}=\frac{\mathrm{c}}{\mathrm{a}}\end{aligned}$
The difference of root can also be found in the same way by manipulating the terms
$\alpha-\beta=\left|\frac{\sqrt{D}}{a}\right|$
Important Results
(i) $\alpha^2+\beta^2=(\alpha+\beta)^2-2 \alpha \beta$
(ii) $\alpha^2-\beta^2=(\alpha+\beta)(\alpha-\beta)$
(iii) $\alpha^3+\beta^3=(\alpha+\beta)^3-3 \alpha \beta(\alpha+\beta)$
(iv) $\alpha^3-\beta^3=(\alpha-\beta)^3+3 \alpha \beta(\alpha-\beta)$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
If and
are roots of the equation,
for some k
and then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
if are the root of equation
,
then the equation has root
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be the roots of the equation
are in A.P. and
then the value of
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If be the ratio of the roots of the quadratic equation in x,
then the least value of m for which
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one real root of the quadratic equation is a cube of the other root, then a value of k is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation, $x^2+(3-\lambda) x+2=\lambda \ {\text {has the least value is: }}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let If
is a root of the quadratic equation,
then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If m is chosen in the quadratic equation such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
are the roots of the quadratic equation,
, then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of the equation
then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of for which sum of squares of roots of
is minimum is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one root of is reciprocal of other, then equation has
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
, then the equation whose roots are
&
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be two real roots of the equation
where,
and
are real numbers. If
then a value of
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If for some $c<0$, the quadratic equation, $2 c x^2-2(2 c-1) x+3 c^2=0$ has two distinct real roots, $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{~b}}$, then the value of the determinant $\left[\begin{array}{ccc}1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c\end{array}\right]$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of the equation
then value of
equals?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the two roots of the equation
.Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of the equation
then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and k>0. If the curve represented by
intersects the y-axis at the point P and Q where PQ=5, then the value of k is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be the roots of
and
be the roots of
. If
forms a geometric progression. Then the ratio
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be in R. If
and
are the roots of the equation,
and
and
are the roots of the equation,
, then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are such that 1 - 2i (here
) is root of
, then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be two real numbers such that
. Let
,
for some integer
. Then, the value of
is ________
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be the roots of the equation
and
form an equilateral triangle with origin. Then, the value of |a| is ____________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one root of is double the other root. Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If sum of roots of equation is -1, then roots of equation are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
, then the equation whose roots are
&
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the sum of the squares of the reciprocals of the roots and
of the equation
is 15, then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be a quadratic polynomial such that
. If one of the roots of
, then the sum of the roots of
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The minimum value of the sum of the squares of the roots of is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be the roots of the quadratic equation
. If
, then
is equal to___________:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of the equation
for each positive integer
then the value of
is equal to__________
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of pairs of real numbers, such that whenever
is a root of the equation
is also a root of this equation, is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let p, q and r be real numbers (p ≠ q, r ≠ 0), such that the roots of the
equation are equal in magnitude but opposite in sign, then
the sum of squares of these roots is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If f (x) is a quadratic expression such that , and −1 is a root of
, then the other root of
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is such that the sum of the cubes of the roots of the equation,
is minimum, then the magnitude of the difference of the roots of this equation is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The quadratic equations x2 - 6x + a = 0 and x2 - cx + 6 = 0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. Then the common root is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
are roots of the equation
, such that
, then p belongs to the set :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
are roots of the equation
If
and
are the two values of
for which the roots
are connected by the relation
then the value of
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the conditions, if the roots of the equation x3 - px2 + qx - r = 0 are in AP.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of the equation $x^2-a x+2 b=0$ are prime numbers, where $\mathrm{a}, \mathrm{b}$ are integers, then $(b-a)$ is equal to $\qquad$ -
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of
the the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of
then
equals.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of
then the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find p if one of the roots of is the square of the other root.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are roots of equation
, then find the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the relation between a and b if one of the roots of the equation is 2 more than the root.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If and
, then the value of
is.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of
, and
, then the value
.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is a root of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a, b \in \mathbf{R}, a \neq 0$ be such that the equation, $a x^2-2 b x+5=0$ has a repeated root $\alpha$, which is also a root of the equation, $x^2-2 b x-10=0$. If $\beta$ is the other root of this equation, then $\alpha^2+\beta^2$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be the roots of equation
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one root of the equation ,
is reciprocal of the other, then,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
is an equation
, With one of roots as
, then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
One root of quadratic equation with rational coefficients is then quadratic will be
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The difference between the corresponding roots of is same and a ≠ b, then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let a, b,c be in arithmetic progression. Let the centroid of the triangle with vertices and
be
If
are the roots of the equation
then the value of
is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of all the roots of the equation $\left|x^{2}-8 x+15\right|-2 x+7=0$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha$ and $\beta$ are distinct real numbers such that $2 \alpha^2=9 \alpha-3$ & $2 \beta^2=9 \beta-3$. If $a_n=2 \alpha^n-\beta^n($ for all $n \in N)$,
then find the value of $\frac{2 a_{2025}+3 a_{2023}}{3 a_{2024}}=?$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta$ be the roots of the equation $x^2-x+2=0$ with $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Then $\alpha^6+\alpha^4+\beta^4-5 \alpha^2$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{6} x+3=0$ such that $\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)$. Let $a$, b be integers not divisible by 3 and $n$ be a natural number such that $\frac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+i b), i=\sqrt{-1}$. Then $n+a+b$ is equal to________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha$ and $\beta$ be the roots of the equation $\mathrm{px}^2+\mathrm{qx}-\mathrm{r}=0$, where $\mathrm{p} \neq 0$. If $\mathrm{p}, \mathrm{q}$ and $\mathrm{r}$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$, then the value of $(\alpha-\beta)^2$ is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be two non-zero real numbers. If
are the roots of the equation
and
are the roots of the equation
, such that
are in A.P, then
is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha$ and $\beta$ are the solutions of the equation $2^{\log z_2(\cos 2 \theta+b \operatorname{cosec} 2 \theta)}=c$, then $\tan (3 \alpha+3 \beta)$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If 2 and 6 are the roots of the equation $a x^2+b x+1=0$, then the quadratic equation, whose roots are $\frac{1}{2 a+b}$ and $\frac{1}{6 a+b}$, is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
An equation has $\frac{1}{2}, 7-3$ is its roots and 2 as the leading coefficient what is the cubic equation
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the product of roots of the equation :
$\frac{1}{x}-\frac{1}{x-2}=3 ; x \neq 0,2$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the sum of roots of the equation:
$x+\frac{1}{x}=3 \quad, \quad x \neq 0$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Form the quadratic equation whose roots are $\frac{2}{3}\: \: and \: \: \frac{7}{4}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of equation $\frac{1}{x+p}+\frac{x}{x+q}=\frac{1}{r}$ are equal in magnitude but opposite in sign, then $(p+q)=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the roots of $(b-c)x^{2}+(c-a)x+(a-b)=0$ are equal then $a+c=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one root of the equation $p x^2+q x+r=0, p \neq 0$ is reciprocal of the other, then,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\mathrm{p}, \mathrm{q}$ are the roots of equation $a^2+b x+c=0$ and $S_n=p^n+q^n$, then, $a S_{n+1}+c S_{n-1}=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the equation $(a-2) x^2-(a-4) x-2=0 \mathrm{}$ has a difference of roots as 3, then the value of a is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha, \beta$ be roots of $x^2+p x+1=0$ and $\gamma, \delta$ are the roots of $x^2+q x+1=0$, then $(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let two numbers have an arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of $b$ for which the sum of the squares of the roots of the equation $x^2-(b-2) x-b-1=0$ assumes the least value is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of a for which one of the roots of $x^2-3 x+2 a=0$ is double of one of the roots of $x^2-x+a=0$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha, \beta, \gamma$ are the roots of $x^3+2 x-9=0$, and
$
\begin{aligned}
& p=\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma} \\
& q=(1-\alpha)(1-\beta)(1-\gamma)
\end{aligned}
$
$r=$ Sum of roots
then $p+q+r{\text { equals }}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $x^{4}-8 x^{3}+a x^{2}+b x+16=0$ has all positive real roots, then the value of 'a' is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $x^2+b x+a c=0$ and $x^2+c x+a b=0$ have a common root, then the other root of the first equation is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha, \beta$ are roots of $a x^2+b x+c=0$ then the equation with roots $2 \alpha, 2 \beta$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta$ be the roots of the equation $x^2+2 \sqrt{2} x-1=0$. The quadratic equation, whose roots are $\alpha^4+\beta^4$ and $\frac{1}{10}\left(\alpha^6+\beta^6\right)$, is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let the number $(22)^{2022}+(2022)^{22}$ leave the remainder $\alpha$ when divided by $3$ and $\beta$ when divided by $7$ . Then $\left(\alpha^2\right.$ $\left.+\beta^2\right)$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If one root of the equation $\small (x-2)(6-x)=c$, is 3 times the other, then '' is equal to?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The relation between p,q and r if $x^{3}+px^{2}+qx+r=0$ has roots which are in A.P.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The roots of the equation $\left|\begin{array}{ccc}a & b & a x+b \\ b & c & b x+c \\ a x+b & b x+c & c\end{array}\right|=0$ are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha, \beta$ are roots of the equation $x^2-4 x+10=0$, then $\ \frac{\alpha}{\beta}+\frac{\beta}{\alpha}$ equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta ; \alpha>\beta$, be the roots of the equation $x^2-\sqrt{2} x-\sqrt{3}=0$. Let $P_n=\alpha^n-\beta^n, n \in N$. Then $(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}$ is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $a, b, c$ be non-zero real roots of the equation $x^3+a x^2+b x+c=0$. Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The product of all solutions of the equation $\mathrm{e}^{5\left(\log _{\mathrm{e}} \mathrm{x}\right)^2+3}=\mathrm{x}^8, \mathrm{x}>0$, is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha, \beta$ be the roots of the equation $x^2-a x-b=0$ with $\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)$. Let $P_n=\alpha^n-\beta^n$. If $P_3=-5 \sqrt{7} i, \quad P_4=-3 \sqrt{7} i, \quad P_5=11 \sqrt{7} i \quad$ and $\mathrm{P}_6=45 \sqrt{7} \mathrm{i}$, then $\left|\alpha^4+\beta^4\right|$ is equal to______________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the squares of all the roots of the equation $x^2+|2 x-3|-4=0$, is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha+i \beta$ and $\gamma+i \delta$ are the roots of $x^2-(3-2 i) x-(2 i-2)=0, i=\sqrt{-1}$, then $\alpha \gamma+\beta \delta$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The quadratic equation $x^2+b x+c=0$ with real coefficients $b$ and $c$, has a complex root $\sqrt{3}-i$ then which of the following represents the value of $c-i b$ ?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha, \beta$ are roots $a x^2+b x+c=0$ of then find the value of $\frac{1}{(a \alpha+b)}+\frac{1}{(a \beta+b)}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $P_{n}=\alpha^{n}+\beta^{n}, n \in \mathbf{N}$. If $P_{10}=123, P_{9}=76$, $\mathrm{P}_{8}=47$ and $\mathrm{P}_{1}=1$, then the quadratic equation having roots $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Consider the equation $\mathrm{x}^2+4 \mathrm{x}-\mathrm{n}=0$, where $\mathrm{n} \in[20,100]$ is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
