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## The Quadratic Equation – Completing The Square Method

Complete The quadratic method is one way to solve the quadratic equation. It’s very simple if you understand how we derived our form method.

Remember that quadratic equations are second degree polynomials and their form can be represented as follows:

Ax^2 + Bx+ C =0

Some quadratics are very simple to solve because they are presented in simple form as follows:

Say (x-3)^2=9

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

i.e. square (x-3)^2 = square (9)

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

By solving the previous equation we will have two answers.

that is, x=3+3 or x=3-3

x=6 or x=0

But what about the situation when our equation is not presented in this form? Most quadratic equations will not square neatly like this. In this case, you first use your math technique to arrange the squares in a well-squared part equal to a number like the example discussed above. Hence, the method of completing the square.

For a typical example:

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

solution

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

4x^2-2x=5 (remember that when you move -5 to the other side of the equation it becomes +5)

Step 2: Divide by the coefficient of your squared X term (which is 4 in our example)

Now the equation becomes:

X^2 – ½X = 5/4

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

½ of -1/2 = -1/4

When you square it, you have 1/16 to add to both sides of the equation which now becomes:

X^2 – 1/2X + 1/16 = 5/4 + 1/16

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

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

Step 5: Find the square root of both sides

x-1/2 = + or – squared (21/16)

solving for x ultimately leads to 2 answers:

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

Congratulations, you have successfully completed the steps to solve a quadratic equation using the quadratic method.

Summary:

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

2. Divide by the coefficient of the term x squared

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

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

Exercise:

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

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