Quadratic equation

An equation like ax²+ bx + c = 0 where a ≠ 0, which contains only one variable and '2' as its highest power is called a quadratic equation. There are two values of the variable in any quadratic equation. The roots of the equations are :

$$ x = \frac {-b \pm \sqrt {b^2 -4ac}}{2a} $$

Quadratic equation are of two types

i. Pure quadratic equation:

A quadratic equation of the form ax²+ c = 0, where the middle term containing power 1 is missed is known as a pure quadratic equation.

e.g. x² = 9 or, x² - 9 = 0
i.e. ax² + c = 0 ( a ≠ o, c = 0 )

ii. Adfected quadratic equation:

The standard form of quardic equation is known as adfected quadratic equation.

ax² + bx + c = 0 is adfected quadratic equation.

e.g. x² - 9x - 15 = 0
i.e. ax² + bx + c = 0 ( a ≠ o, b ≠ o)

Solution of quadratic equation

Quadratic equation is a second-degree equation of one variable. So we get two solutions of the variable contained by this equation. The solution of a quadratic equation is known as the roots. Hence, a quadratic equation has two roots. The solution of roots which are obtained from the quadratic equation should satisfy the equation. There are three major methods for solving quadratic equation, which are

1. Factorization method
2. Completing square method
3. Using formula

a) Solving a quadratic equation by factorization method:

In this method, the quadratic equation ax² + bx+ c = 0 is factorized and expressed as the product of two linear factors. Again, the two linear factors are also solved to get the solution of the equation. the roots that is obtained from the equation should satisfy the given quadratic equation.

b) Solving a quadratic equation by completing square method:

We transpose the constant term (c) to the R.H.S and the remaining parts ax² + bx is expressed in the perfect square expression.

c) Solving a quadratic equation by using formula:

In this method, we have obtained two roots as \(\frac{-b+\sqrt{b^2-4ac}}{2a}\) and \(\frac{-b-\sqrt{b^2-4ac}}{2a}\)

the two roots of the quadratic equation ax²+ bx+ c = 0 can be obtained using a direct formula,

x = \(\frac{-b±\sqrt{b^2-4ac}}{2a}\)

'a' is the coefficient of x²

'b' is the coefficient of x and

'c' is the constant term.

While solving any quadratic equation using the formula, first we simplify and bring the equation in the simplest form. Then we compare with ax²+ bx+ c = 0 and find a, b and c. Finally, we use x = \(\frac{-b±\sqrt{b^2-4ac}}{2a}\) and find two roots of the given quadratic equation.