Quadratic Equation - Practice Questions & MCQ

An expression of the form  f(x) = a₀xⁿ + a₁x^n-1 + a₂x^n-2 +... + a_n-1x + a_n , is called a polynomial expression.

Where x is variable and a₀, a₁, a₂, …….. , a_n  are constant, known as coefficients and  a₀ ≠ 0, n is non-negative integer, 

Degree: The highest power of variable the polynomial expression is called the degree of the polynomial. In , the highest power of x is n, so the degree of this polynomial is n. 

If coefficients are real numbers then it is called real polynomial, and when they are complex numbers, then the polynomial is called complex polynomial.

Root of polynomial: 

If f(x) is a polynomial, then f(x) = 0 is called 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 = α is a root of the equation f(x) = 0, then f(α) = 0.

Eg, x=2 is a root of x² - 3x + 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 quadratic equation.

Standard form of quadratic equation is ax² + bx + c = 0

Where a, b and c are constants (they may be real or imaginary) and called the coefficients of the equation and  (a is also called the leading coefficient).

Eg, -5x² - 3x + 2 = 0, x² = 0, (1 + i)x² - 3x + 2i = 0

As 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:

Where D is called the discriminant of the quadratic equation, given by  ,

Proof: