-
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
Quadratic Equation - (Concept)
An expression of the form $f(x)=a_0 x^n+a_1 x^{n-1}+a_2 x^{n-2}+\ldots+a_{n-1} x+a_n$, is called a polynomial expression.
Where x is variable and $a_0, a_1, a_2, \ldots \ldots, a_n$ are constant, known as coefficients and $a_0 \neq 0, n$ is non-negative integer,
Degree: The highest power of the variable in the polynomial expression is called the degree of the polynomial. In $a_0 \cdot x^n+a_1 \cdot x^{n-1}+\ldots+a_n$ , the highest power of x is n, so the degree of this polynomial is n.
If coefficients are real numbers then it is called a real polynomial, and when they are complex numbers, then the polynomial is called a complex polynomial.
The root of a polynomial:
If $f(x)$ is a polynomial, then $f(x)=0$ is called a polynomial equation.
The value of $x$ for which the polynomial equation, $f(x)=0$ is satisfied is called a root of the polynomial equation.
If $x=\alpha$ is a root of the equation $f(x)=0$, then $f(\alpha)=0$.
$\mathrm{Eg}, \mathrm{x}=2$ is a root of $\mathrm{x}^2-3 \mathrm{x}+2=0$, as $x=2$ satisfies this equation.
A polynomial equation of degree n has n roots (real or imaginary).
Quadratic equation:
A polynomial equation in which the highest degree of a variable term is 2 is called a quadratic equation.
Standard form of quadratic equation is $a x^2+b x+c=0$
Where $\mathrm{a}, \mathrm{b}$ and c are constants (they may be real or imaginary) and called the coefficients of the equation and $a \neq 0$ (a is also called the leading coefficient).
$
E g,-5 x^2-3 x+2=0, x^2=0,(1+i) x^2-3 x+2 i=0
$
As the degree of quadratic polynomial is 2, so it always has 2 roots (number of real roots + number of imaginary roots $=2$ )
The root of the quadratic equation is given by the formula:
$
\begin{aligned}
& x=\frac{-b \pm \sqrt{D}}{2 a} \\
& \text { or } \\
& x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}
\end{aligned}
$
Where D is called the discriminant of the quadratic equation, given by $D=b^2-4 a c$,
Proof:
$
a x^2+b x+c=0
$
Take, 'a' common
$
\begin{aligned}
& a\left(x^2+\frac{b}{a} x+\frac{c}{a}\right)=0 \\
& a\left[\left(x+\frac{b}{2 a}\right)^2-\frac{b^2}{4 a^2}+\frac{c}{a}\right]=0 \\
& \left(x+\frac{b}{2 a}\right)^2=\frac{b^2-4 a c}{4 a^2} \\
& \left(x+\frac{b}{2 a}\right)= \pm \frac{\sqrt{b^2-4 a c}}{2 a} \\
& x=-\frac{b}{2 a} \pm \frac{\sqrt{b^2-4 a c}}{2 a} \\
& x=\frac{-b \pm \sqrt{b^2-4 a c}}{2 a}
\end{aligned}
$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
If $\alpha$ and $\beta$ are the roots of the equation $\frac{1}{i Z}-i Z=2(\sin \theta-i \cos \theta)$ where $0<\theta<\pi$ and $i=\sqrt{-1}$, and z is complex number, then the value of $|\alpha-i|+|\beta-i|$ is $2\sqrt n$. Find the value of $n$.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Value of for which
represents a quadratic equation with real coefficients is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Roots of equation are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the roots of the equation, ,is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the roots of the equation
then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let Z be the set of integers. If and
then the number of subsets of the set
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. |
|
All x satisfying the inequality
, lie in the interval:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be the roots of the quadratic equation
Then
is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If are the length of the sides of a triangle, then r cannot be equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha$ and $\beta$ be two roots of the equations $x^2+2 x+2=0$, then $\alpha^{15}+\beta^{15}$ is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $B=\left[\begin{array}{ccc}5 & 2 \alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1\end{array}\right]$ is the inverse of a $3 \times 3$ matrix A, then the sum of all values of $\alpha$ for which $\operatorname{det}(A)+1=0$, is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the solutions of the equation is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If be the roots of the equation
, then the least value of n for which
is:
| 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. |
|
Let a,b, and c be in G.P. with common ratio r, where and
. If 3a, 7b, and 15c are the first three terms of an A.P., then the 4th term of this A.P. is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let is a root of
then
equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of all the real values of x satisfying the equation is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The roots of equation are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If has discriminant 49, then a equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Consider the quadratic equation $(c-5) x^2-2 c x+(c-4)=0, c \neq 5$. Let $S$ be the set of all integral values of c for which one root of the equation lies in the interval $(0,2)$ and its other root lies in the interval $(2,3)$. Then the number of the element in S is::
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^{x}+1=0$ is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be the roots of the equation,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The product of the roots of the equation is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be the roots of
. If
for
, then the value of
is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If , the number of solutions of the given equation when
is _______.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If the arithmetic mean and geometric mean of the pth and qth terms of the sequence -16,8,-4,2..... satisfy the equation , then p+q is equal to ________
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be defined by
.
Let be given as
. Then , the sum of all values of x for which
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of
$
3+\frac{1}{4+\frac{1}{3+\frac{1}{4+\frac{1}{3+\ldots . . \infty}}}}
$
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following is quadratic expression for all
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What is the value of a for which represents a quadratic equation with real coefficients?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of all real values of x satisfying the equation is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let p(x) be a quadratic polynomial such that p(0)=1. If p(x) leaves remainder 4 when divided by (x−1) and it leaves remainder 6 when divided by (x+1); then :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the cubes of all the roots of the equation ______________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
be two real numbers such that
and
. Then
is equal to.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be the roots of the equation
Then, the value of
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of distinct real roots of the equation
is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
For $\mathrm{p}, \mathrm{q} \in \mathbf{R}$, consider the real valued function $\mathrm{f}(\mathrm{x})=(\mathrm{x}-\mathrm{p})^2-\mathrm{q}, \mathrm{x} \in \mathbf{R}$ and $\mathrm{q}>0$. Let $\mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3$ and $\mathrm{a}_4$ be in an arithmetic progression with mean P and positive common difference. If $\left|\mathrm{f}\left(\mathrm{a}_{\mathrm{i}}\right)\right|=500$ for all $\mathrm{i}=1,2,3,4$, then the absolute difference between the roots of $\mathrm{f}(\mathrm{x})=0$ is $\qquad$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of all real values of for which
is equal to ___________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be such that
and
then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let be two roots of the equation
. Then
is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of all integral values of for which the equation
in x has no real roots, is______
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Consider two quadratic expressions and
,
such that their discriminants are equal. If
has a root
, then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Domain of is :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of real roots of the equation :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Solutions of is are:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Domain of is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Domain of is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is an identity in x. Then the value of
equals.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is an identity in x, then the value of a are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is an identity in a, then the value that x can take is/are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is an identity in x, then the value of x can be.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The roots of the equation are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Solution (s) of is /are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Solution (s) of is / are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Solution of is / are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The solution (s) of is/are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Solution (s) of is/are.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is a prime number ,then value of x may be:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If there are two equations
,
such that
then:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If then the equation
have:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If is a root of
then
equals:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of for which
becomes a quadratic identity equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the number of solutions of $ \frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-c)(x-a)}{(b-c)(b-a)} + \frac{(x-a)(x-b)}{(c-a)(c-b)} = x $,
where $ a \ne b \ne c $.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the number of solutions of
$ \frac{(x+a)^2}{(a-b)(a-c)} + \frac{(x+b)^2}{(b-c)(b-a)} + \frac{(x+c)^2}{(c-a)(c-b)} = x^2 $, where $ a \ne b \ne c $.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The equation denotes the greatest integer function, has
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let and
Then
is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the number of values of 'c' in $2 x^2-7 x+c=0$, such that, $x^2+5 x-4=0$ has one common root with the given equation:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $(1+p)$ is a root of quad. eq. $x^{2}-px+(1+p)=0$
then its roots are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If 3 is a root of $x^{2}+kx-24=0$ ,it is also a root of
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha$ is a root of $2 x^2-3 x-1=0$, then $-\alpha^3-4 \alpha^2+3 \alpha-2=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
What are the roots of quadratic equation: $x^{2}+2x-143=0$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following is a quadratic equation in x with real co-efficients?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which of the following is quadratic expression ?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Which one of the following is quadratic equation with complex coefficients?
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$\operatorname{Cos} \theta$ is a root of the equation $25 x^2+5 x-12=0,-1<x<0$, then the value of $\operatorname{Sin} 2 \theta$ is,
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of real roots of the equation, $e^{4 x}+e^{3 x}-4 e^{2 x}+e^x+1=0$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the separate equations of the lines represented by the equation $x^2-2 y^2+x y+x-y=0$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $(a-1) x^2+(b+2) x+c^2=0$ is satisfied by $x=0,4,-10$. Then the value of $a+b+c$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $P(x)$ is a polynomial of degree $\leq 2$ such that $P(0)=0, P(1)=1$ and $\mathrm{P}^{\prime}(\mathrm{x})>0 \forall \mathrm{x} \in[0,1]$, then S , the set of such polynomials is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $a+\frac{1}{a}=4$, then the value of $a^2+\frac{1}{a^2}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $a+\frac{1}{a}=4$, then $a^3+\frac{1}{a^3}$ equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of $\sqrt{4-4x+x^2}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Roots of $x^{2}+\left ( 2-i \right )x-2i=0$ are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The Solution of $\left ( log_{2}\: x \right )^{2}-3\: log_{2}\: x+2=0$ is/are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The solution of equation $9^{x}-10.3^{x}+9=0$ is/are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function
$
f(x)=\left(x-a_1\right)\left(x-a_2\right)+\left(x-a_2\right)\left(x-a_3\right)+\left(x-a_3\right)\left(x-a_1\right)
$
with $a_1, a_2, a_3 \varepsilon \mathbb{R}$. Then $f(x) \geq 0$ if and only if
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\log _3\left(\log _2\left(x^2+x+2\right)\right)=1$, the the values of $x$ are 2 and ____:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\mathrm{p}, \mathrm{q}, \mathrm{r}$, are real, $p \neq q$, then the roots of the equation $(p-q) x^2-5(p+q) x-2(p-q)=0$ are
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of value of x for the equation $\small x=\sqrt{12+\sqrt{12+\sqrt{12+...\infty}}}$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Value of a if $(x-a)$ is a factor of $x^4-a x^3-4 a x+16=0$, is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $P$ be a non-zero polynomial such that $P(1+x)=P(1-x)$ for all real $x$, and $P(1)=0$. et $m$ be the largest integer such that $m(x-1)^m$ divides $P(x)$ for all such $P(x)$ Then $m$ equals
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Suppose $a, b, c$ are real nimber, and each of the equations $x^2+2 a x+b^2=0$ and $x^2+2 b x+c^2=0$ has two distinct real roots. Then the equation $x^2+2 c x+a^2=0$ has
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of solution pairs $(x,y)$of the simultaneous equations $log_{\frac{1}{3}}(x+y)+log_{3}(x-y)=2$
$2^{{y}^{2}}=512^{x+1}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The product of all the rational roots of the equation $\left(x^2-9 x+11\right)^2-(x-4)(x-5)=3$, is equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $f: \mathbf{R}-\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree 2, satisfying $f(\mathrm{x}) f\left(\frac{1}{\mathrm{x}}\right)=f(\mathrm{x})+f\left(\frac{1}{\mathrm{x}}\right)$. If $f(\mathrm{~K})=-2 \mathrm{~K}$, then the sum of squares of all possible values of K is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $S=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}$. Then $\sum_{\mathrm{x} \in \mathrm{S}}(2 \mathrm{x}-1)^2$ is equal to____________.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The value of $p$ for which $\left(p^2-3 p+2\right) x^2+p x+\left(p^2-p\right)=0$ has more than 2 roots is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Find the entire truth set of $x^4 -1 = 0$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3 x}-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3 x-1=0$. If $P_n=\alpha^n+\beta^n$ and $Q_n=\gamma^n+\delta^n$, then $\frac{\mathrm{P}_{25}+\sqrt{3} \mathrm{P}_{24}}{2 \mathrm{P}_{23}}+\frac{\mathrm{Q}_{25}-\mathrm{Q}_{23}}{\mathrm{Q}_{24}}$ is equal to
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The sum of the squares of the roots of $|x-2|^2+|x-2|-2=0$ and the squares of the roots of $x^2-2|x-3|-5=0$, is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
We are already familiar with the quadratic equations and have solved them in the set of real numbers in the cases where discriminant is non-negative, i.e., 0,
