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The Quadratic Equation – Completing The Square Method
Complete The method of squares is a way to solve a quadratic equation. It’s so simple if you understand how we derived our formula method.
Remember that quadratic equations are quadratic polynomials and their form can be represented as follows:
Ax^2 + Bx+ C =0
Some quadratics are very easy to solve because they come in a simple form like below:
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 a square root of a number, say 9 for example, the result would be either + 0r – )
Solving for x in the equation above, we are going to have two answers.
i.e. x=3+3 or x=3-3
x=6 or x=0
But what about the situation where our equation does not come in this form. Most quadratic equations won’t square perfectly like this. In this case, you first use your mathematical technique to arrange the quadratics in a perfectly square part equal to a number as in the example discussed above. Thus, the completion of the square method.
For a typical example:
Solve the quadratic equation 4x^2 -2x-5=0
The solution
Step 1: move -5 to the RHS to 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 term X squared (which is 4 in our example)
The equation now becomes:
X^2 – ½X = 5/4
Step 3: Take half the coefficient of term X, square it and add it to both sides
½ of -1/2 = -1/4
When you square it, 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 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 – sqrt(21/16)
solving for x finally leads to 2 answers:
X=1/2- sqrt(21/16) or X= ½ + sqrt(21/16)
Congratulations, you have successfully completed the steps for solving a quadratic equation using the method of squares.
Summary:
1. Move the number part to the right of the equation
2. Divide by the coefficient of the term x squared
3. Take half the coefficient of the term x, square it and add it to both sides of the equation
4. Rearrange your 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
Practice:
Solve X^2 +6X-7=0 by completing the square method
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