The Quadratic Equation – Complete the Square Method

Complete The square method is a way to solve quadratic equations. It’s so simple if you understand how we derive our formula method.

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

Ax^2 + Bx+ C =0

Some quadratics are very simple to solve because they come in a simple form like the following:

Say (x-3)^2=9

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

i.e. 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 would be + 0r – )

Solving for x in the above equation we are going to get two answers.

that is, x=3+3 or x=3-3
x=6 or x=0

But what about the situation when our equation doesn’t come in this form? Most quadratic equations will not balance perfectly this way. In this case, first use his mathematical technique to organize the quadratics into a perfectly square 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 Right Side of the Equation (Right Side of Right Side)

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

Step 2 – Divide by the coefficient of your X squared 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, square it, and add it to both sides

½ of -1/2 = -1/4

When you square it, you have to add 1/16 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 has a simple square shape 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 using the method of completing the square.

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 your equation by putting the right side into square form 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