Lesson 5
How Many Solutions?
 Let’s use graphs to investigate quadratic equations that have two solutions, one solution, or no solutions.
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.
 \(3x^2 = 81\)
 \((x1)(x+1) 9 = 5x\)
 \(x^2 9x + 10 = 32\)
 \(6x(x8) = 29\)
Problem 2

Here are equations that define quadratic functions \(f, g\), and \(h\). Sketch a graph, by hand or using technology, that represents each equation.
\(f(x)=x^2+4\)
\(g(x) = x(x+3)\)
\(h(x)=(x1)^2\)
 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 \((x5)^2=0\). 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.
For which algae population is the area decreasing more rapidly? Explain how you know.
Problem 5
If the equation \((x4)(x+6)=0\) is true, which is also true according to the zero product property?
only \(x  4 = 0\)
only \(x + 6 = 0\)
\(x  4 = 0\) or \(x + 6 = 0\)
\(x=\text4\) or \(x=6\)
Problem 6
 Solve the equation \(25=4z^2\).
 Show that your solution or solutions are correct.
Problem 7
To solve the quadratic equation \(3(x4)^2 = 27\), Andre and Clare wrote the following:
Andre
\(\displaystyle \begin {align} 3(x4)^2 &= 27 \\ (x4)^2 &= 9 \\ x^2  4^2 &= 9 \\ x^2  16 &= 9 \\ x^2 &= 25 \\ x = 5 \quad &\text{ or }\quad x = \text 5\\ \end {align}\)
Clare
\(\displaystyle \begin{align} 3(x4)^2 &= 27\\ (x4)^2 &= 9\\ x4 &= 3\\ x &= 7\\ \end{align}\)
 Identify the mistake each student made.
 Solve the equation and show your reasoning.
Problem 8
Decide if each equation has 0, 1, or 2 solutions and explain how you know.
 \(x^2 144=0\)
 \(x^2 +144=0\)
 \(x(x5)=0\)
 \((x8)^2=0\)
 \((x+3)(x+7)=0\)