Quadratic Equation - Practice Questions & MCQ

An expression of the form $f(x)=a_0 x^n+a_1 x^{n-1}+a_2 x^{n-2}+\ldots+a_{n-1} x+a_n$, 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}
$