IM Alg1.7.5 Lesson: How Many Solutions?
Decide whether each statement is true or false. Be prepared to explain your reasoning.
3 is the only solution to .
A solution to is .
has two solutions.
and are the solutions to .
Han is solving three equations by graphing.
What are the solutions?
Then, use the graph to solve the equation. Be prepared to explain how you use the graph for solving.
Solve the third equation using Han’s strategy: (x-5)(x-3) = -4
Think about the strategy you used and the solutions you found. Why might it be helpful to rearrange each equation to equal 0 on one side and then graph the expression on the non-zero side?
How many solutions does each of the three equations have?
and
and
Analyze the graphs and explain how each pair helps to solve the related equation.
Use the graphs to help you find a few other equations of the form that have whole-number solutions.
Find a pattern in the values of that give whole-number solutions.
Without solving, determine if and have whole-number solutions. Explain your reasoning.
x²=121.
x²-31=5.
(x-4)(x-4)=0.
(x+3)(x-1)=5.
(x+1)²=-4.
(x-4)(x-1)=990.
Consider . Priya reasons that if this is true, then either or . So, the solutions to the original equation are 12 and 6. Do you agree? If not, where was the mistake in Priya’s reasoning?
Consider . Diego says to solve we can just divide each side by to get , so the solution is 10. Mai says, “I wrote the expression on the left in factored form, which gives , and ended up with two solutions: 0 and 10.” Do you agree with either strategy? Explain your reasoning.