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So far we have solved some quadratic equations by factoring and by the Square Root Property (i.e. taking the square root both of sides). However, not all quadratic equations are factorable so we can not always depend on factoring.

Also, to use the Square Root Property, we needed to have the x squared on the LHS, or a binomial squared on the LHS.

In this section we will learn another method called completing the square. This method will give us the power to solve any quadratic equation. The trick to this section is getting the binomial squared on the LHS. Let’s observe the square of a binomial on the LHS.

2.

1.

Note that on the RHS of the trinomial, the last term is ½ of the coefficient of the middle term squared.

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Solving Quadratic Equations by Completing the Square (C.T.S.)

Procedure: To solve a quadratic equation of the form ax2+bx+c=0 by C.T.S.

Your Turn Problem #1

1. Move the constant to the RHS, leave a space after the bx term.

2. Divide by ‘a’ on both sides to get x2 on the LHS. In this example, it is not necessary.

3. Multiply ‘b’ (the the coefficient of the x-term) by ½, then square the result. Note: This number will always be positive since any real number squared is positive. Add this number to both sides.

4. The LHS is a perfect square trinomial. It can be factored and written as the square of a binomial. Also simplify the RHS.

5. Then, solve by using the Square Root Property.

+ 9 +9

Your Turn Problem #2 (C.T.S.)

Move the constant to the RHS, take ½ of the middle number and add it to both sides.

Solution:

+ 25 +25

Factor the LHS and write it as a binomial squared. Simplify the RHS.

Now use the square root property.

Solution: (C.T.S.)

Your Turn Problem #3

Move the constant to the RHS, take ½ of the middle number and square it. Add this result to both sides.

+ 1 +1

Factor the LHS and write it as a binomial squared. Simplify the RHS.

Now use the square root property.

Solution: (C.T.S.)

Your Turn Problem #4

Since it is not a complex number, the solution is written as a single fraction.

Move the constant to the RHS, take ½ of the middle number and square it. Add this result to both sides. Fractions are fine. Leave improper and don’t convert to decimal.

Factor the LHS and write it as a binomial squared. Simplify the RHS.

Now use the square root property.

Solution: (C.T.S.)

3

3

3

Your Turn Problem #5

Since it is not a complex number, the solution is written as a single fraction.

Move the constant to the RHS. Then, divide by 3 to get the x2 on the LHS.

Take ½ of the middle number and square it. Add this result to both sides

Factor the LHS and write it as a binomial squared. Simplify the RHS.

Now use the square root property.

The End.

5-30-07

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