Quadratic Equation

Introduction

A quadratic equation is an equation of the form ax² + bx + c = 0 where a, b, and c are constants and a ≠ 0.

The origin of word "quadratic" is Latin. In Latin, "quadratic" is used for "square". Since the highest power of the unknown variable that appears in the equation is square(power 2), hence similar equations came to be known as quadratic equation.

The standard form of a quadratic equation is ax² + bx + c = 0, where

◾ a, b and c are constants and

◾ a is not equal to 0 (zero).

Here are some of the examples of standard form of a quadratic equation:

Equation	Coefficient
x2 + 2x + 1 = 0	where a = 1, b = 2 and c = 1
x2 + 5x + 11 = 0	where a = 1, b = 5 and c = 11

◾ When b is not equal to zero

Some of the examples of complete form of a quadratic equation
◾ x2 + 2x + 1 = 0 where a = 1, b = 2 and c = 1
◾ x2 + 5x + 11 = 0 where a = 1, b = 5 and c = 11

◾ When b is equal to zero, the equation is known as pure or incomplete quadratic equation in x.

Some of the examples of Pure or Incomplete form of a quadratic equation
◾ 6x2 – 24 = 0 where b = 0
◾ 6x2 – 11 = 0 where b = 0

The Roots or solution of the quadratic equation of ax² + bx + c =0 are the values of the variable 'x' which satisfy the quadratic equation i.e. make the ax²+bx+c equals to zero.

There are three methods for solving a quadratic equation:

◾ By doing factorization

◾ By completing the square

◾ By using the quadratic formula

Step by step process to solve the quadratic equation by factorization method:

Step 1: Rearrange the equation in standard form which is ax² + bx + c = 0. If right hand side is not zero, take it to left hand side and make the right hand side zero.

Step 2: Fully factorise the left hand side.

Step 3: Use the Null Factor law: If pq = 0 then p = 0 or q = 0.

Step 4: Solve the resulting linear equations by equating each of the linear factor to zero. These values of x will be the solution of the quadratic equation.

Pitfall:

You may be tempted to divide both sides by an expression involving x. If you do this then you only get one solution of equation (one value or one root of x) and may lose the other solution (value of x).

As you would be aware by now that all quadratic equations cannot be factorised easily. For example, x² + 4x + 1 cannot be factorised by simple factorisation. It means we cannot write x² + 4x + 1 in the form (x - a)(x - b) where a, b are rational numbers.

There is an alternate way to solve equations like x² + 4x + 1 = 0 that is by completing the square.

Equations of the form ax² + bx + c = 0 can be converted to the form (x + p)² = q . It is easy to find the solutions in that way.

Step by step process to solve the quadratic equation by completing the square:

Step I: Arrange the quadratic equation in standard form of ax² + bx + c = 0.

Step II: now, divide both sides of the equation by the co-efficient of x² if it is not already 1.

Step III: Shift the constant term to the right hand side.

Step IV: Add the square of one-half of the co-efficient of x to L.H.S. and R.H.S.

Step V: Write the L.H.S as complete square and simplify the R.H.S.

Step VI: Solve for x by taking the square root of L.H.S. and R.H.S.

Let us solve quadratic equation -3x² + 12x + 5 = 0 by "completing the square"

The quadratic formula, which also may be used to solve any quadratic equation, results from solving the quadratic equation ax² + bx + c = 0 , a ≠ 0 for x by completing the square.

There are certain cases when solving a quadratic equation by factorisation or completing the square is time consuming, lengthy or difficult. In such cases, we use the quadratic formula to solve the quadratic equation.

Step by step process to solve the quadratic equation by quadratic formula:

Step I: Arrange the Quadratic Equation in the standard form of ax² + b x + c = 0.

Step II: Compare the quadratic equation which is to be solved with the standard quadratic equation and find out the values of coefficients a, b, and c.

Step III: Put these values of a, b, and c in Quadratic formula.

As you can notice in the formula. Quadratic formula calculates two values of x: x₁ and x₂, where

These two values of x for which ax² + bx + c = 0 holds true, are called solutions of Quadratic Equation, also called roots of Quadratic Equation.

Discriminant

We have learnt that the quadratic formula is

Roots(x1, x2) =	−b ± √   b2 − 4ac
	2a

In the quadratic formula above, the quantity "b² - 4ac" which is under the square root sign is called the discriminant of the quadratic equation.

Discriminant = b² - 4ac

The expression "b² – 4ac" tells about the nature of the roots of the quadratic equation. The roots may be real, equal or imaginary.

There are three possible cases:

◾ If b² – 4ac < 0, then roots will be imaginary and unequal.

◾ If b² – 4ac = 0, then roots will be real, equal and rational . (This means the left hand side of the equation is a perfect square).

◾ If b² – 4ac > 0, then the roots are real and unequal.

If b² – 4ac > 0, then the roots are real and unequal and their are two possibilities - Here the roots could be rational or irrational

◾ b² – 4ac is a perfect square, the roots are real, rational and unequal . (This mean the equation can be solved by the factorization).

◾ b² – 4ac is a not perfect then roots are real, irrational and unequal.

Summary

Discriminant value Cases	Roots of quadratic	Factorisation of quadratic
Discriminant value > 0	two real distinct roots	two distinct linear factors
Discriminant value = 0	two identical real roots	two identical linear factors
Discriminant value < 0	No real roots	Unable to factorise