Completing the Square

Solving by factoring and extracting the square roots are techniques that only work for very special quadratic equations. In this section, we will develop a method that will solve any quadratic equation. This technique will be used to develop the quadratic formula.

Solve by Completing the Square Playlist

If a quadratic expression

is to be a perfect square then the constant term must equal 16. In this case, we can see that

To know 16 was the number needed to complete the square we divided our middle term B = 8 by 2 to get 4 then squared that to obtain 16.

Complete the square then factor.

The leading coefficient must be 1 if this method is to work. If it is not equal to 1 we could always factor the coefficient out first as in the next examples.

Determine the constant that completes the square.

Now that you know how to complete the square for a quadratic expression you are ready to solve any quadratic equation. Notice that the next example does not factor, so we will solve it by completing the square.

Solve by completing the square:

   Step 1: Add or subtract the constant term to the other side.

   Step 2: Add (b/2)^2 to both sides. Here B = −10 so (−10/2)^2 = (−5)^2 = 25

   Step 3: Factor and then apply the square root property.

   Step 4: Solve as usual.

Tip: Be sure to add the “magic number,”

to both sides so that the equality is retained.

Solve by extracting roots.

So far we have seen examples where the leading coefficients are 1. But if this technique is to be useful for solving quadratic equations of the form

we must address the case where a is a number other than 1.

Solve by completing the square.

Extra Step: Factor out the leading coefficient.

Here B = −6 so (B/2)^2 = (−6/2)^2 = (−3)^2 = 9. Caution! Adding 9 inside the parentheses is equivalent to adding 18 here.

Solve.

Video Examples on YouTube:

Solve by Extracting Square Roots

So far we have been able to solve quadratic equations using the factoring method.

The idea was to set the equation to zero, factor it, then set each factor to zero and solve. The zero factor property allowed us to solve factorable quadratic equations.

Solve by Factoring Playlist

Solve by factoring.

At this point we will learn a new technique that works for quadratic equations that are in the following form:

The idea here is to isolate the x^2 term and then apply the square root property. For example, if

then it is fairly clear that x = -3 or x = 3 which can be gotten using the following property:

When applying the square root to both sides of an equation remember to include the + or -.

Solve by Extracting Roots Playlist

Solve by extracting roots:

   Step 1: Isolate the square term.

   Step 2: Apply the square root property.

   Step 3: Be sure to include the + or - and solve as usual.

Compare the next three problems to the first three of this section, they are the same questions. Notice that the two very different techniques give us the same solutions.

Instructional Video: Solving Quadratic Equations Using Square Roots

Solve by extracting roots.

Solve.

Video Examples on YouTube: