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Isabel is a researcher studying whooping crane populations in a particular area. She has determined that, due to natural fluctuations between predators and the whooping cranes, the population can be modelled by the following function, where p is the population after x years:

 

p(x) = −x4 + 12x3 − 37x2 + 6x + 101

 

She has determined that this function will be valid for the next 7 yr.

 

 

Isabel would like to determine when the population will be 45 whooping cranes. One way to solve this problem is graphically, by drawing a horizontal line at 45 and determining where the line crosses the population curve.

 

Based on this graph, it appears that the population will be 45 after 2, 4, and 7 yr.

 

 

However, Isabel remembered how much fun she had learning about polynomials in Mathematics 30-1, so she decided to solve the problem algebraically. This corresponds to solving the equation p(x) = 45.

 

−x4 + 12x3 − 37x2 + 6x + 101 = 45

 

As a first step in learning how to solve this equation, complete Try This 1.

 

Try This 1

 

Solve the following quadratic equations. Make note of any similarities and differences in how you solved the equations.

1. w2 − 3w − 10 = 0
2. w2 + w − 9 = 3

Save your responses in your course folder.

 

Share 1

 

With a partner or in a group, share your solutions from Try This 1. How was your first step in solving the first equation different from the first step in solving the second equation?

 

If required, save a record of your discussion in your course folder.