The Quadratic Equation – Completing the Square Method

Complete The square method is a way to solve quadratic equations. It’s so simple if you understand how we derived 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 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 does not come in this form? Most quadratic equations will not square perfectly this way. In this case, first use his mathematical technique to arrange 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 that when you move -5 to the other side of the equation it becomes +5)

Step 2: Divide by the coefficient of your X term squared (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 eventually 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 completing the square method.

Summary:

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

2. Divide by the coefficient of the x term 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 making the right side square 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