Completing the Square

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
Standards: HSA-SSE.B.3HSG-GPE.A.1
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15 questions

Completing the Square

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Fill in the Box

For each expression, what value would need to be in the box in order for the expression to be a perfect square trinomial?

1) x2+10x+3x^2 + 10x + \boxed{\phantom{3}}

2) x216x+3x^2 - 16x + \boxed{\phantom{3}}

3) x2+40x+3x^2 + 40x + \boxed{\phantom{3}}

4) x2+5x+3x^2 + 5x + \boxed{\phantom{3}}

Complete the Process

Here is the equation of a circle: x2+y26x20y+105=0x^2 + y^2 - 6x - 20y + 105 = 0.

Elena wants to find the center and radius of the circle. Here is what she’s done so far.

Step 1: x26x+y220y=105x^2 - 6x + y^2 - 20y = -105

Step 2: x26x+9+y220y+100=105+9+100x^2 - 6x + 9 + y^2 - 20y + 100 = -105 + 9 + 100

Step 3: x26x+9+y220y+100=4x^2 - 6x + 9 + y^2 - 20y + 100 = 4

5) What did Elena do in the first step?

6) Why did Elena add 9 and 100 to the left\textit{left} side of the equation in Step 2?

7) Why did Elena add 9 and 100 to the right\textit{right} side of the equation in Step 2?

8) What should Elena do next?

9) What is the center of this circle?

10) What is the radius of this circle?

11) Draw a graph of the circle.

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Your Turn

Here is the equation of a circle: x2+y22x+4y4=0x^2 + y^2 - 2x + 4y - 4 = 0.

12) What is the center of the circle?

13) What is the radius of the circle?

14) Explain your reasoning.

15) Draw a graph of the circle.

A template for answering this question. Ask your instructor for an alternative.