5.12: Completing the Square When the Leading Coefficient Doesn't Equal 1
The area of another parallelogram is given by the equation , where x is the length of the base. What is the length of this base?
Guidance
When there is a number in front of , it will make completing the square a little more complicated. See how the steps change in Example A.
Example A
Determine the number c that completes the square of .
Solution: In the previous concept, we just added , but that was when . Now that , we have to take the value of a into consideration. Let's pull out the GCF of 2 and 8 first.
Now, there is no number in front of .
.
Add this number inside the parenthesis and distribute the 2.
So, .
Example B
Solve
Solution:
1. Write the polynomial so that and are on the left side of the equation and the constants on the right.
2. Pull out from everything on the left side. Even if is not divisible by , the coefficient of needs to be 1 in order to complete the square.
3. Now, complete the square. Determine what number would make a perfect square trinomial.
To do this, divide the term by 2 and square that number, or .
4. Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add to keep the equation balanced.
5. Factor the left side and simplify the right.
6. Solve by using square roots.
Be careful with the addition of step 2 and the changes made to step 4. A very common mistake is to add to both sides, without multiplying by for the right side.
Example C
Solve .
Solution: Let’s follow the steps from Example B.
1. Write the polynomial so that and are on the left side of the equation and the constants on the right.
2. Pull out from everything on the left side.
3. Now, complete the square. Find .
4. Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add to keep the equation balanced.
5. Factor the left side and simplify the right.
6. Solve by using square roots.
Intro Problem Revisit We can't factor , so let's follow the step-by-step process we learned in this lesson.
1. Write the polynomial so that and are on the left side of the equation and the constants on the right.
2. Pull out from everything on the left side.
3. Now, complete the square. Find .
4. Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add to keep the equation balanced.
5. Factor the left side and simplify the right.
6. Solve by using square roots.
However, because x is the length of the parallelogram's base, it must have a positive value. Only results in a positive value, so the length of the base is .
Guided Practice
Solve the following quadratic equations by completing the square.
1.
2.
Answers
Use the steps from the examples above to solve for .
1.
2.
Explore More
Solve the quadratic equations by completing the square.
Solve the following equations by factoring, using square roots, or completing the square.
Problems 13-15 build off of each other.
- Challenge Complete the square for . Follow the steps from Examples A and B. Your final answer should be in terms of and .
- For the equation , use the formula you found in #13 to solve for .
- Is the equation in #14 factorable? If so, factor and solve it.
- Error Analysis Examine the worked out problem below.
Plug the answers into the original equation to see if they work. If not, find the error and correct it.
Image Attributions
Description
Learning Objectives
Here you'll learn how to complete the square for quadratic equations in standard form.