8.1–Solving quadratic equations by completing the square A ...

8.1–Solving quadratic equations by completing the square
A quadratic equation is an equation in this form: . We’ve dealt with
quadratic equations before and we’ve even solved them. For example the equation
can be factored into . Now setting the two factors
equal to zero and solving for x gives us . What if the quadratic equation
won’t factor? For example . We’ll get back to that later.
Let’s factor the expression . This factors into or .
What about the expression ? It factors into or .
These two trinomials are called perfect square trinomials.
Notice something about the 2 nd and 3 rd terms of each. If you take half of the 2 nd term and square
it, you get the 3 rd term, right? This is not a coincidence. What number must be added to
to make it a perfect square trinomial? Well, half of �2 is �1 and �1 squared is 1. So
is a perfect square trinomial. What about ? Take half of 3 and get ,
now square and get . So is a perfect square trinomial.
Now back to quadratic equations that won’t factor. There are 4 ways to solve quadratic
equations:
1) Square root method
2) Factoring (if possible)
3) Completing the square
4) Quadratic formula
Factoring doesn’t always work but completing the square and the quadratic formula always work.
These are the steps for completing the square:
1) Divide by a.
2) Move c.
3) Take half of b, square it, and add it to both sides of the equation.
4) Write the left side as a perfect square.
5) Take the square root of both sides of the equation and solve.

Ex. 1
Solve by completing the square:
1. *Recall the standard form of a quadratic equation and label
each term . Now let’s complete the
square. First, divide by a: . That
leaves us with . Second, move c:
. Third, take half of b (half of 4 is 2), square
it (2 squared is 4), and add it to both sides:
. Fourth, write the left side as a
perfect square (in other words, factor it): .
Fifth, take the square root of both sides of the equation and
solve: , . Since you
can’t take the square root of 10 evenly, leave it as a radical
and don’t forget the ± in front of it (you always need that
when taking the square root of a number). Now isolate the
x and get .