Lesson 14

Completing the Square (Part 3)

  • Let’s complete the square for some more complicated expressions.

Problem 1

Select all expressions that are perfect squares.

A:

\(9x^2 + 24x + 16\)

B:

\(2x^2 + 20x + 100\)

C:

\((7 - 3x)^2\)

D:

\((5x + 4)(5x - 4)\)

E:

\((1 - 2x)(\text- 2x + 1)\)

F:

\(4x^2 + 6x + \frac94\)

Problem 2

Find the missing number that makes the expression a perfect square. Next, write the expression in factored form.

  1. \(49x^2 - \underline{\hspace{.5in}} x + 16\)
  2. \(36x^2 + \underline{\hspace{.5in}} x + 4\)
  3. \(4x^2 - \underline{\hspace{.5in}} x + 25\)
  4. \(9x^2 + \underline{\hspace{.5in}} x + 9\)
  5. \(121x^2 + \underline{\hspace{.5in}} x + 9\)

Problem 3

Find the missing number that makes the expression a perfect square. Next, write the expression in factored form.

  1. \(9x^2 + 42x + \underline{\hspace{.5in}}\)
  2. \(49x^2 - 28x +\underline{\hspace{.5in}}\)
  3. \(25x^2 + 110x + \underline{\hspace{.5in}}\)
  4. \(64x^2 - 144x +\underline{\hspace{.5in}}\)
  5. \(4x^2 + 24x + \underline{\hspace{.5in}}\)

Problem 4

  1. Find the value of \(c\) to make the expression a perfect square. Then, write an equivalent expression in factored form.
    standard form \(ax^2+bx+c\) factored form \((kx+m)^2\)
    \(4x^2+4x\)  
    \(25x^2-30x\)  
  2. Solve each equation by completing the square.

    \(4x^2+4x=3\)

    \(25x^2-30x+8=0\)

Problem 5

For each function \(f\), decide if the equation \(f(x)=0\) has 0, 1, or 2 solutions. Explain how you know.

A

Parabola facing up with 2, x intercepts.

B

Parabola facing up with vertex on x axis.

C

Parabola facing up with 0, x intercepts.

D

Parabola facing down with 2, x intercepts.

E

Function on a grid. X axis from negative 4 to 5, by 1’s. Y axis from negative 12 to 4, by 2’s. Origin, O. Parabola opens downward with vertex at 1 comma 0.

F

Function on a grid. X axis from negative 10 to 8, by 2’s. Y axis from negative 28 to 4, by 8’s. Origin, O. Parabola opens downward with a vertex at 0 comma negative 8.
(From Unit 7, Lesson 5.)

Problem 6

Solve each equation.

\(p^2+10=7p\)

\(x^2+11x+27=3\)

\((y+2)(y+6)=\text-3\)

 

 

(From Unit 7, Lesson 9.)

Problem 7

Which function could represent the height in meters of an object thrown upwards from a height of 25 meters above the ground \(t\) seconds after being launched?

A:

\(f(t)=\text-5t^2\)

B:

\(f(t)=\text-5t^2+25\)

C:

\(f(t)=\text-5t^2+25t+50\)

D:

\(f(t)=\text-5t^2+50t+25\)

(From Unit 6, Lesson 6.)

Problem 8

A group of children are guessing the number of pebbles in a glass jar. The guesses and the guessing errors are plotted on a coordinate plane.

horizontal axis, guess. scale 0 to 32, by 4's. vertical axis, absolute guessing error. scale 0 to 12, by 2's. 
  1. Which guess is furthest away from the actual number?
  2. How far is the furthest guess away from the actual number?
(From Unit 4, Lesson 13.)