Lesson 12

Completing the Square (Part 1)

  • Let’s learn a new method for solving quadratic equations.

Problem 1

Add the number that would make the expression a perfect square. Next, write an equivalent expression in factored form.

  1. \(x^2 - 6x\)
  2. \(x^2 + 2x\)
  3. \(x^2 + 14x\)
  4. \(x^2 - 4x\)
  5. \(x^2 + 24x\)

Problem 2

Mai is solving the equation \(x^2 + 12x = 13\). She writes:

\(\displaystyle \begin{align} x^2 + 12x &= 13\\ (x + 6)^2 &= 49\\ x &= 1 \text { or } x = \text- 13\\ \end{align}\\\)

Jada looks at Mai’s work and is confused. She doesn’t see how Mai got her answer.

Complete Mai’s missing steps to help Jada see how Mai solved the equation.

 

Problem 3

Match each equation to an equivalent equation with a perfect square on one side.

A:

\(x^2 + 8x = 2\)

B:

\(x^2 + 10x = \text- 13\)

C:

\(x^2 - 14x = 5\)

D:

\(x^2 + 2x = 0\)

E:

\(x^2 + 4x - 5 = 0\)

F:

\(x^2 - 20x = \text- 9\)

1:

\((x-7)^2 = 54\)

2:

\((x+5)^2 = 12\)

3:

\((x-10)^2 = 91\)

4:

\((x+4)^2 = 18\)

5:

\((x+1)^2 = 1\)

6:

\((x+2)^2 = 9\)

Problem 4

Solve each equation by completing the square.

\(x^2-6x+5=12\)

\(x^2-2x=8\)

\(11=x^2+4x-1\)

\(x^2-18x+60=\text-21\)

Problem 5

Rewrite each expression in standard form.

  1. \((x+3)(x-3)\)
  2. \((7+x)(x-7)\)
  3. \((2x-5)(2x+5)\)
  4. \((x+\frac18)(x-\frac18)\)
(From Unit 7, Lesson 8.)

Problem 6

To find the product \(203 \boldcdot 197\) without a calculator, Priya wrote \((200+3)(200-3)\). Very quickly, and without writing anything else, she arrived at 39,991. Explain how writing the two factors as a sum and a difference may have helped Priya.

(From Unit 7, Lesson 8.)

Problem 7

A basketball is dropped from the roof of a building and its height in feet is modeled by the function \(h\).

Here is a graph representing \(h\).

Select all the true statements about this situation.

Curve on grid. Horizontal axis, time in seconds, 0 to 2. Vertical axis, height in feet, 0 to 60. Curve starts at 0 comma 50. Decreases down and to the right. Hits horizontal axis near 1 point 7 5.
A:

When \(t=0\) the height is 0 feet.

B:

The basketball falls at a constant speed.

C:

The expression that defines \(h\) is linear.

D:

The expression that defines \(h\) is quadratic.

E:

When \(t=0\) the ball is about 50 feet above the ground.

F:

The basketball lands on the ground about 1.75 seconds after it is dropped.

(From Unit 6, Lesson 5.)

Problem 8

A group of students are guessing the number of paper clips in a small box.

The guesses and the guessing errors are plotted on a coordinate plane.

What is the actual number of paper clips in the box?​​​​​​

horizontal axis, guess. scale 0 to 32, by 4's. vertical axis, absolute guessing error. scale 0 to 12, by 2's. 

​​​​​

(From Unit 4, Lesson 13.)