Completing the Square

Diving into the world of completing the square might feel like embarking on a heroic quest in the realm of math! Fear not, brave mathematician, for this step-by-step guide is your trusty map through the maze of numbers and variables. We’re about to turn completing the square into a mathematical adventure that’ll make your brain do somersaults of joy!

Completing the Square Steps

Get ready to sprinkle some magic on quadratic equations! 🧙‍♂️✨ This step-by-step guide will turn the algebraic journey into a thrilling adventure!

1. Make sure the equation is in this format: y = ax² + bx + c
2. Subtract c from both sides.
3. Factor out a.
4. Find (b/2)².
5. Add that to left and right sides of the equation. (If a is a number other than 1, you have to multiply your answer to part 4 by a when adding it to the left side.)
6. Simplify the left side by combining like terms.
7. Factor the right side.
8. Subtract or add the number from the left side, so it is on the right side.

Example 1

y – 10x = x² + 4

1. y = x² + 10x + 4
2. y – 4 = x² + 10x
3. Skip because a is 1 in this case.
4. (10/2)² = 25
5. 25 + y – 4 = x² + 10x + 25
6. y + 21 = x² + 10x + 25
7. y + 21 = (x + 5)²
8. y = (x + 5)² – 21

Example 2

y = -16x² + 96x + 3

1. y = -16x² + 96x + 3
2. y – 3 = -16x² + 96x
3. y – 3 = -16(x² – 6x)
4. (-6/2)² = 9
5. -16(9) + y – 3 = -16(x² -6x + 9)
6. y – 147 = -16(x² -6x + 9)
7. y – 147 = -16(x – 3)²
8. y = -16(x – 3)² + 147

.

Use Completing the Square to Graph a Parabola

Get ready to transform your graphing game from drab to fab with Completing the Square! 🚀✨ Grab your math wand (or pencil, whichever feels more magical), and let’s sprinkle some graphing magic together! 🎩🌈

Let’s use the example Equation y = -2² -4x -1.

Part A: Complete the square.

1. y = -2² -4x -1
2. y + 1 = -2² -4x
3. y + 1 = -2(x² + 2x)
4. (2/2)² = 1
5. -2(1) + y + 1 = -2(x² + 2x +1)
6. y – 1 = -2(x² + 2x +1)
7. y – 1 = -2(x+1)²
8. y = -2(x+1)² + 1

Part B: Find the vertex.

1. vertex = (h, k) when y = a(x – h)² + k
2. y = -2(x+1)² + 1
3. h = -1 and k = 1
4. vertex = (-1, 1)

Part C: Plot Points.

1. Plot the vertex (-1, 1).
2. Plug values for x into the original equation: y = -2x² -4x -1
3. 0 is the easiest number to plug in: y = -2(0)² -4(0) -1 = -1 → (0,-1)
4. 1 is also easy to plug in: y = -2(1)² -4(1) -1 = -7 → (1, -7)
5. Plot the points these three points: (-1, 1), (0, -1), and (1, -7).
6. Use the axis of symmetry which runs up and down through the vertex to plot the mirror image of (0, -1) and (1, -7).

.

Send your questions about this page and suggestions on how to improve it via email to Marci@RegalLessons.com or through carrier pigeon.

.

More Math Resources

Now that you’ve learned about completing the square, click here for a math-tastic adventure by unleashing the power of our more awesome math study tools! 🚀🔢✨ We have pages just like this one to help you with the graphing transformations, finding asymptotes, and so much more!

.

Book a private tutor!

Guess what! We have math tutors that can help you with completing the square and other math challenges. If you live in the South Bay (of Los Angeles) we can provide at-your-home tutoring. If you live outside the area, we’re happy to work with you virtually. To see our availability and schedule a tutoring session, please use the booking plug-in below!

• Regal Tutors accepts bookings up to 80 days in advance.
• If you see a window that says “Your phone ____ is already associated with another email.” Click “UPDATE.”
• Click here to watch a video on how to schedule a tutor. 

.

Let’s be friends!

• 
• 
• 
• 
• 
•