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## The Quadratic Equation – Completing The Square Method

The completing square method is a way to solve a quadratic equation. It is very simple if you understand how we derived our formula method.

Remember that quadratic equations are polynomials of second degree order and can be represented in the form:

Ax^2 + Bx + C = 0

Some quadratics are very simple to solve because they come in simple forms like below:

Say (x-3)^2=9

This type of quadratic equation can be solved quickly by taking the square root of both sides of the equation.

ie sqrt ( x-3)^2 =sqrt(9)

x-3=+0r-3 (note that when you take the square root of a number say 9 for example, the result will be either + 0r – )

By solving for x in the above equation we will have two answers.

ie x=3+3 or x=3-3

x=6 or x=0

But, what happens if our equation does not come in this form. Most quadratic equations do not square neatly in this way. In this case you can first use your mathematical techniques to arrange the squares of the same number into a neat square like the example above. Thus, completing the class method.

For a typical example:

Solve the quadratic equation 4x^2 -2x-5=0

solution

Step 1: Move -5 to the RHS equation (RHS-to the right)

4x^2-2x=5 (note that moving -5 to the other side of the equation makes it +5)

Step 2: Divide your X by the co-efficient of the square term (which is 4 in our example)

The equation now becomes:

X^2 – ½X = 5/4

Step 3: Take half the coefficient of the X term and square it and add it to both sides

-1/2 of ½ =-1/4

When you square this you have 1/16 added to both sides of the equation which now becomes:

X^2 – 1/2X + 1/16 = 5/4 + 1/16

Step 4: Convert the left hand side to a square form and simplify the RHS

(x-1/2)^2 = 21/16 (now you have a simple quadratic form like our first example)

Step 5: Find the square root of both sides

x-1/2 = + or – sqrt(21/16)

Solving for x ultimately leads to 2 answers:

X=1/2- sqrt(21/16) or X= ½ + sqrt(21/16)

Congratulations you have successfully completed the steps to solve a quadratic equation by completing the quadratic method.

Summary:

1. Move the number part to the right hand side of the equation

2. Divide x squared by the coefficient of the term

3. Take half the coefficient of the x term, square it and add it to both sides of the equation

4. Rearrange your equation by squaring the right-hand side and simplifying the left-hand side. Take the square root of both sides, remembering the + or – sign on the right. Finally solve for the two possible values of X

Exercise:

Solve X^2 +6X-7=0 by completing the square method

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