Lesson 5

How Many Solutions?

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Problem 1

Rewrite each equation so that the expression on one side could be graphed and the \(x\)-intercepts of the graph would show the solutions to the equation.

  1. \(3x^2 = 81\)
  2. \((x-1)(x+1) -9 = 5x\)
  3. \(x^2 -9x + 10 = 32\)
  4. \(6x(x-8) = 29\)

Problem 2

  1. Here are equations that define quadratic functions \(f, g\), and \(h\). Sketch a graph, by hand or using technology, that represents each equation.

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

    Blank coordinate grid, origin O.

    \(g(x) = x(x+3)\)

    Blank coordinate grid, origin O.

    \(h(x)=(x-1)^2\)

    Blank coordinate grid, origin O.
  2. Determine how many solutions each \(f(x)=0, g(x)=0\), and \(h(x)=0\) has. Explain how you know.

Problem 3

Mai is solving the equation \((x-5)^2=0\). She writes that the solutions are \(x=5\) and \(x=\text- 5\). Han looks at her work and disagrees. He says that only \(x=5\) is a solution. Who do you agree with? Explain your reasoning.

Problem 4

The graph shows the number of square meters, \(A\), covered by algae in a lake \(w\) weeks after it was first measured.

In a second lake, the number of square meters, \(B\), covered by algae is defined by the equation \(B = 975 \boldcdot \left(\frac{2}{5}\right)^w\), where \(w\) is the number of weeks since it was first measured.

Graph of function.

For which algae population is the area decreasing more rapidly? Explain how you know.

(From Algebra1, Unit 5, Lesson 6.)

Problem 5

If the equation \((x-4)(x+6)=0\) is true, which is also true according to the zero product property?

A:

only \(x - 4 = 0\)

B:

only \(x + 6 = 0\)

C:

\(x - 4 = 0\) or \(x + 6 = 0\)

D:

\(x=\text-4\) or \(x=6\)

(From Algebra1, Unit 7, Lesson 4.)

Problem 6

  1. Solve the equation \(25=4z^2\).
  2. Show that your solution or solutions are correct.
(From Algebra1, Unit 7, Lesson 3.)

Problem 7

To solve the quadratic equation \(3(x-4)^2 = 27\), Andre and Clare wrote the following:

Andre

\(\displaystyle \begin {align} 3(x-4)^2 &= 27 \\ (x-4)^2 &= 9 \\ x^2 - 4^2 &= 9 \\ x^2 - 16 &= 9 \\ x^2 &= 25 \\ x = 5 \quad &\text{ or }\quad x = \text- 5\\ \end {align}\)

Clare

\(\displaystyle \begin{align} 3(x-4)^2 &= 27\\ (x-4)^2 &= 9\\ x-4 &= 3\\ x &= 7\\ \end{align}\)

  1. Identify the mistake each student made.
  2. Solve the equation and show your reasoning.
(From Algebra1, Unit 7, Lesson 3.)

Problem 8

Decide if each equation has 0, 1, or 2 solutions and explain how you know.

  1. \(x^2 -144=0\)
  2. \(x^2 +144=0\)
  3. \(x(x-5)=0\)
  4. \((x-8)^2=0\)
  5. \((x+3)(x+7)=0\)