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!

Written Steps

1. Make sure the equation is in this format: y = ax² + bx + c
2. Subtract c from both sides.
3. If a is a number other than 1, factor it out.
4. Find (b/2)².
5. This is where it can get tricky.
   • If a = 1: add your answer from Step 4 to the left and right sides of the equation.
   • If a ≠ 1: add your answer from Step 4 times a to the left side of the equation and add your answer from Step 4 to the right side of the equation inside the parenthesis.
6. Simplify the left side by combining like terms.
7. Factor the right side to form something like this a (x + #)².
8. Subtract or add the number from the left side, so it is on the right side.

Example for a = 1

y – 10x = x² + 4

1. Make sure the equation is in this format: y = ax² + bx + c
   • y = x² + 10x + 4
2. Subtract c from both sides.
   • y – 4 = x² + 10x
3. Skip because a is 1.
4. Find (b/2)².
   • (10/2)² = 25
5. Add your answer from Step 4 to the left and right sides of the equation.
   • 25 + y – 4 = x² + 10x + 25
6. Simplify the left side by combining like terms.
   • y + 21 = x² + 10x + 25
7. Factor the right side to form something like this a (x + #)².
   • y + 21 = (x + 5)²
8. Subtract or add the number from the left side, so it is on the right side.
   • y = (x + 5)² – 21

Example for a ≠ 1

y = -16x² + 96x + 3

1. Make sure the equation is in this format: y = ax² + bx + c
   • y = -16x² + 96x + 3
2. Subtract c from both sides.
   • y – 3 = -16x² + 96x
3. Factor out a.
   • y – 3 = -16(x² – 6x)
4. Find (b/2)².
   • (-6/2)² = 9
5. Add your answer from Step 4 times a to the left side of the equation and add your answer from Step 4 to the right side of the equation inside the parenthesis.
   • -16(9) + y – 3 = -16(x² -6x + 9)
6. Simplify the left side by combining like terms.
   • y – 147 = -16(x² -6x + 9)
7. Factor the right side to form something like this a (x + #)².
   • y – 147 = -16(x – 3)²
8. Subtract or add the number from the left side, so it is on the right side.
   • y = -16(x – 3)² + 147

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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).

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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!

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