# Quadratic Equations

If a_{0}, a

_{1}, a

_{2}, â€¦, a

_{n}are real numbers and 'n' is a non-negative integer, then a function p(x) = (a

_{0}+ a

_{1}x + a

_{2}x

^{2}+ a

_{3}x

^{3}+â€¦+ a

_{n}x

^{n}) is called a polynomial in x over reals. The value of 'n' in a given polynomial is called the degree of the polynomial.

The most general form of quadratic equation is ax^{2 }+ bx + c = 0 where a, b and c are real constants, a â‰ 0 and x is a real variable.

**Example :** 3x^{2 }+ 5x + 9 = 0 , x^{2 }+ 9 = 0, 2x^{2 }+ 5x + 7= 0, x^{2 }- x + 2 = 0 and = 0

**Roots of a Quadratic Equation**

If the two numbers Î± and Î² are the two roots of the quadratic equation, p(x) = ax^{2 }+ bx + c = 0, then Î± and Î² are called the roots of the quadratic equation p(x).

For x^{2 }- x - 6 = 0, the roots are 3 and (-2)

**Note**

If ax

^{2}+ bx + c becomes 0 for x = Î± then Î± is known as the root of the quadratic equation

Which of the following are quadratic equations?

(a) 2x^{2 }+ 5x + 3 = 0 (b) 7x + 5 = 0

(c) (d) x^{3 }+ 1 = 0

(e) 2x^{2 }+ 5x + 3 = 2x^{2 }+ 4x + 5.

(a) Quadratic equation

(b) Linear equation

(c) Quadratic equation

(d) It is not a quadratic equation.

(e) Transforming R. H. S. to L. H. S., we get

2x^{2 }+ 5x + 3 - 2x^{2 }- 4x - 5 = 0

â‡’ x - 2 = 0

.^{.}. It is a linear equation.

For the quadratic equation find out which of the following are solutions

(a) x = (b) x = (c) x = .

(a) Substituting x = on L.H.S. of the equation we get,

L.H.S. â‰ R.H.S.

Therefore, x = is not a solution.

(b) Substituting x = on the L.H.S.,

= = R.H.S.

Therefore, x = is a solution of the equation.

(c) Substituting x = on the L.H.S. we get,

= = R.H.S.

Therefore, x= - is also a solution of the equation.