Completing the Square

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Created by Illustrative MathematicsIllustrative Mathematics, CC BY 4.0
Subject: Geometry
Grade: 9-12
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24 questions

Completing the Square

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Problem 1

1) Suppose a classmate missed the lessons on completing the square to find the center and radius of a circle. Explain the process to them. If it helps, use a problem you’ve already done as an example.

Problem 2

Complete the square for each of the following trinomials. Choose your answers from the numbers in this set: 16, 20.25, 64, 100.

2) x28x+3 x^2 - 8x + \boxed{\phantom{3}}

3) x2+20x+3 x^2 + 20x + \boxed{\phantom{3}}

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

5) x2+9x+3x^2 + 9x + \boxed{\phantom{3}}

Problem 3

Here is the equation of a circle: x2+y2+4x10y+20=0x^2 + y^2 + 4x - 10y + 20 = 0.

6) What is the center of the circle?

7) What is the radius of the circle?

Problem 4

8) Select all\textbf{all} the expressions that can be factored into a squared binomial. Write each corresponding letter in the answer box and separate letters with commas.

a) y2+2y+1y^2 + 2y + 1 \quad\quad b) w2+5w+254w^2 + 5w + \frac{25}{4} \quad\quad c) y210y+5y^2 - 10y + 5 \quad\quad d) x210x+25x^2 - 10x + 25 \quad\quad e) x2+10x+25x^2 + 10x + 25 \quad\quad f) w2+20w+40w^2 + 20w + 40

Problem 5

9) An equation of a circle is given by (x+3)2+(y9)2=52(x+3)^2 + (y-9)^2 = 5^2. Apply the distributive property to the squared binomials and rearrange the equation so that one side is 00.

Problem 6

Here is the equation of a circle: (x+1)2+(y3)2=16(x + 1)^2 + (y - 3)^2 = 16.

10) Graph the circle.

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

Find the distance from the center of the circle to each point on the list.

11) (2,1)(2, 1)

12) (4,1)(4, 1)

13) (3,3)(3, 3)

Based on these distances, state whether each point is inside, on, or outside the circle?

14) (2,1)(2,1)

15) (4,1)(4,1)

16) (3,3)(3,3)

Problem 7

17) The triangle whose vertices are (3,1)(3,-1), (2,4)(2,4) and (5,1)(5,1) is transformed by the rule (x,y)(2x,5y)(x,y) \rightarrow (2x,5y) . Is the image similar or congruent to the original figure?

a)

The image is congruent to the original triangle.

b)

The image is similar but not congruent to the original triangle.

c)

The image is neither similar nor congruent to the original triangle.

Problem 8

A cube has side length 3 inches and a sphere has a radius of 3 inches.

18) Before doing any calculations, predict which solid has greater surface area to volume ratio.

a)

cube

b)

sphere

c)

the surface area to volume ratios are the same

19) Calculate the surface area of the cube.

20) Calculate the volume of the cube.

21) Calculate the surface area to volume ratio of the cube.

22) Calculate the surface area of the sphere.

23) Calculate the volume of the sphere.

24) Calculate the surface area to volume ratio of the sphere.