Lesson 5

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. \(6x(x-8) = 29\)

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

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

   

   

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

   

   

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

   

   

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

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.

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.

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

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

only \(x - 4 = 0\)

only \(x + 6 = 0\)

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

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

1. Solve the equation \(25=4z^2\).
2. Show that your solution or solutions are correct.

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

1. Identify the mistake each student made.
2. Solve the equation and show your reasoning.

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. \((x+3)(x+7)=0\)