Lesson 6

Completing the Square

Problem 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.

Solution

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Problem 2

Match each expression with the value needed in the box in order for the expression to be a perfect square trinomial.

Solution

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Problem 3

Find the center and radius of the circle represented by the equation $$x^2+y^2+4x-10y+20=0$$.

Solution

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Problem 4

Select all the expressions that can be factored into a squared binomial.

A:

$$y^2+2y+1$$

B:

$$w^2+5w+\frac{25}{4}$$

C:

$$y^2-10y+5$$

D:

$$x^2-10x+25$$

E:

$$x^2+10x+25$$

F:

$$w^2+20w+40$$

Solution

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(From Unit 6, Lesson 5.)

Problem 5

An equation of a circle is given by $$(x+3)^2+(y-9)^2=5^2$$. Apply the distributive property to the squared binomials and rearrange the equation so that one side is 0.

Solution

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(From Unit 6, Lesson 5.)

Problem 6

1. Graph the circle $$(x+1)^2+(y-3)^2=16$$.
2. Find the distance from the center of the circle to each point on the list.
1. $$(2,1)$$
2. $$(4,1)$$
3. $$(3, 3)$$
3. What do these distances tell you about whether each point is inside, on, or outside the circle?

Solution

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(From Unit 6, Lesson 4.)

Problem 7

The triangle whose vertices are $$(3,\text-1), (2,4),$$ and $$(5,1)$$ is transformed by the rule $$(x,y) \rightarrow (2x,5y)$$. 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.

Solution

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(From Unit 6, Lesson 3.)

Problem 8

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

1. Before doing any calculations, predict which solid has greater surface area to volume ratio.
2. Calculate the surface area, volume, and surface area to volume ratio for each solid.

Solution

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(From Unit 5, Lesson 16.)