Quadratic Equation (3)

This time I will discuss the quadratic formula. This formula actually derived using the completing square method. In spite of writing all the steps, we just pick the final result.

Suppose we want to find the roots of the quadratic equation

STEP ONE : First move the constant to the right hand side of the equation.

STEP TWO : Divide both sides with a. We have this form of equation,

STEP THREE : Then we add both sides with the square of half of the x’s coeffiecient

The left hand sides is a complete square form.

STEP FOUR : Then we can rewrite is as a square of something, have a look.

STEP FIVE : Take the square root of both sides. Simplify the right hand side.

STEP SIX : Isolate the x to find the roots of the equation.

The final result is known as the quadratic formula. I have some examples and exercises. You can download them by click the following link.

Quadratic Equation, using quadratic formula.

Quadratic Equation (2)

We have discuss the method of factorization to find the roots of a quadratic equation. Now we will study the completing square method.

As depicted by its name, this procedure use a complete square form. Suppose we want to find the roots of the quadratic equation

STEP ONE : First move the constant to the right hand side of the equation.

STEP TWO : Divide both sides with 2. We have this form of equation

Note : 2 is the coefficient of x square or a. You may have another value of a. Just divide both sides with a.

STEP THREE : Then we add both sides with the square of half of the x’s coefficient. In this example, the coefficient of x is 2. One half of 2 is 1. The square of 1 is still 1. So we add both sides with 1 to have the following equation.

The left hand sides is a complete square form.

STEP FOUR : Then we can rewrite is as a square of something, have a look.

STEP FIVE : Take the square root of both sides.

STEP SIX : Isolate the x to find the roots of the equation.

This is how the completing square method works. This method is stronger than the factorization method. But it is a little bit tedious. We can use this method to find the roots of all quadratic equations. If a equal to one, then we can skip step two.

I have some examples and exercises. You can download them by click the following link.

Quadratic Equation, the completing square method.

Quadratic Equation (1)

A quadratic equation has a general form of

… (1)

The values of x’s that satisfies the equation are called the roots.

There are three common ways to find the roots, they are

1. Factorization
2. Completing square
3. Quadratic formula

In this posting I will discuss the first method, that is the factorization method.

This is the simplest method, yet the weakest one. This is particularly easy when a = 1. To find the roots, imagine the following illustration.

We have to find two numbers p and q that their product equal to a. We also have to find two numbers m and n that their product equal to c. Moreover, p.n + q.m = b. If there are such numbers, then the quadratic equation can be factorized ast

Therefore the roots of the quadratic equation are :

I have some examples and exercises. You can download them by click the following link.

Quadratic Equation, the factorization method.