Section 009 ) Quadratic equation

Introduction

A quadratic polynomial is a polynomial of degree 2. It is of the form $ax^2+bx+c$ where a,b,c are constants and $a\ne0$. this is so because if a = 0 then the polynomial does not a have a degree 2 but 1 and it becomes a linear polynomial.

A quadratic equation is of the form  $ax^2+bx+c=0$ where $a,b,c$ are constants and $a\ne0$.

Solution of quadratic equation

Completing the square

Completing the square method for solvling quadratic method is to convert the quadratic equation of the form $ax^2+bx+c=0$ to the form $(x+m)^2=c$ and then solve the same.

Following steps are taken to solve the quadratic equation.

1) Take the constant to the RHS.
So we get

$ax^2+bx=-c$

2) Divide both sides of the equation by a to make the coefficient of $x^2$ 1.

We get $x^2+\frac{b}{a}x = \frac{-c}{a}$

To make the LHS a perfect square we need to add  $(\frac{b}{2a})^2$ to the LHS and hence we need to add the same on the RHS .

We get  $x^2+\frac{b}{a}x + (\frac{b}{2a})^2 = \frac{-c}{a} + (\frac{b}{2a})^2 $

Or  $(x+\frac{b}{a})^2 = \frac{-c}{a} + (\frac{b}{2a})^2 $

Then taking the square root on both sides we can solve the same.

Let us take a couple of examples to illustrate the same.

Example 1

Solve $x^2+6x+8=0$

Take 8 to the RHS that is $x^2+6x= - 8$

As coefficient of $x^2$  is 1 we do not have anything to do in this step

To complete the square  add $(\frac{6}{2})^2$ or 9 on both sides to get $x^2+6x+9 = 9-8 =1$

Or $(x+3)^2 = 1$

Or $(x+3) = \pm 1$

Or $x = -3 \pm 1$ giving 2 solutions x = -4 or -2

Example 2

Solve $x^2+3x-4=0$

Take -4 to the RHS that is $x^2+3x= 4$

As coefficient of $x^2$  is 1 we do not have anything to do in this step

To complete the square  add $(\frac{3}{2})^2$ or $\frac{9}{4}$ on both sides to get $x^2+3x+\frac{9}{4} = 4 + \frac{9}{4}$

Or $(x+\frac{3}{2})^2 = \frac{25}{4}$

Or $(x+\frac{3}{2}) = \pm \frac{5}{2}$

Or $x = \frac{-3}{2}  \pm \frac{5}{2}$ giving 2 solutions x = -4 or 1

The other methods are

1) By facrorisation

2) Quadratic Formula

We shall be discussing the methods subseqently