Quadratic Equation - Practice Questions & MCQ

An expression of the form $f(x)=a_0{x}^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\dots +a_{n-1}x+a_n$, is called a polynomial expression.

Where x is variable and $a_0,a_1,a_2,\dots \dots ,a_n$ are constant, known as coefficients and $a_0\ne 0,n$ is non-negative integer,

Degree: The highest power of the variable in the polynomial expression is called the degree of the polynomial. In $a_0\cdot x^n+a_1\cdot x^{n-1}+\dots +a_n$ , 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 $a{x}^2+bx+c=0$
Where $\mathrm{a},\mathrm{b}$ and c are constants (they may be real or imaginary) and called the coefficients of the equation and $a\ne 0$ (a is also called the leading coefficient).

$Eg,-5x^2-3x+2=0,x^2=0,(1+i)x^2-3x+2i=0$
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:

$\begin{array}{rl}& x=\frac{-b\pm \sqrt{D}}{2a}\\ & \text{ or }\\ & x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\end{array}$
Where D is called the discriminant of the quadratic equation, given by $D=b^2-4ac$,

Proof: 

$a{x}^2+bx+c=0$
Take, 'a' common

$\begin{array}{rl}& a(x^2+\frac{b}{a}x+\frac{c}{a})=0\\ & a[{(x+\frac{b}{2a})}^2-\frac{b^2}{4a^2}+\frac{c}{a}]=0\\ & {(x+\frac{b}{2a})}^2=\frac{b^2-4ac}{4a^2}\\ & (x+\frac{b}{2a})=\pm \frac{\sqrt{b^2-4ac}}{2a}\\ & x=-\frac{b}{2a}\pm \frac{\sqrt{b^2-4ac}}{2a}\\ & x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\end{array}$