Name: ________________________________________________Date: _______________________Hour: ___________

Unit 4: Solving Quadratic Functions Review Key 1. Which of the questions below results in a different answer than the other three? Circle the question that is different and explain how you know. Solve the equation (2𝑥 + 3)(𝑥 − 4) = 0.

Find the values of x for which 2𝑥 + 3 = 0 or 𝑥 − 4 = 0.

Find the values of x for which (2𝑥 + 3) + (𝑥 − 4) = 0.

Find the roots of the equation. (2𝑥 + 3)(𝑥 − 4) = 0

The first one can be solved by doing 2𝑥 + 3 = 0 and 𝑥 − 4 = 0. That is the exact same as the second one. The last one is asking for the roots. That is the same thing as asking when the equation equals 0, so it can be solved by doing 2𝑥 + 3 = 0 and 𝑥 − 4 = 0 too. The other equation has a + between it, and that changes things, so that is the different one.

2. Write an example of a quadratic function in standard form with roots that are 3 and 7. If the roots are 3 and 7, then 3 and 7 make the equation equal 0. That means that it has factors (x-3) and (x-7) because those factors will be 0 when x=3 and when x=7. So the equation must be (𝑥 − 3)(𝑥 − 7). Now we need to write it in standard form. We need to simplify by multiplying the two polynomials. (𝑥 − 3)(𝑥 − 7) Multiply everything in one polynomial by everything in the other polynomial. 𝑥 2 − 7𝑥 − 3𝑥 + 21 Combine like terms. 𝑥 2 − 10𝑥 + 21 Done!

3. A student factored the quadratic function below as shown. Is their work correct? Explain. No, this is not correct.

𝑥 2 + 12𝑥 + 8 = (𝑥 + 2)(𝑥 + 4)

They got the 2 and the 4 because those numbers multiply to give us 8, but they were also supposed to find two numbers that add to give us 12. 2 and 4 don’t add to 12, so this is not the correct factored form of the quadratic function.

4. Use the graph below to fill in each blank in the equation with the symbol + or – to complete the equation that models the graph shown. Explain how you knew what symbols to put.

𝑦 = (𝑥

-

9)(𝑥

+

4)

The graph is further to the right than to the left, so that means 9 will make the function 0 and -4 will also make the function 0 because it must cross the x-axis at positive 9 and negative 4. If 9 makes the function 0, then (𝑥 − 9) must be one of the factors. That means there must be a – in the first box. If -4 makes the function 0, then (𝑥 + 4) must be one of the factors. That means there must be a + in the second box.

Name: ________________________________________________Date: _______________________Hour: ___________ 5. Fill in the blank on the quadratic expression below so that the expression is factorable. Explain how you knew what could go there. 2

𝑥 +

𝑥 − 18

In order for this to be factorable, there must be two numbers that multiply to -18 and add to the middle term. Here are the things that multiply to -18 and what they add to: -1 and 18 adds to 17 1 and -18 adds to -17 -2 and 9 adds to 7 2 and -9 adds to -7 -3 and 6 adds to 3 3 and -6 adds to -3 So if the middle term is 17, -17, 7, -7, 3, or -3 then the expression can be written in factored form because then there are two numbers that multiply to -18 and add to any of the numbers shown. Any of those answers is fine. 2

6. Which of the graphs below represents the function 𝑔(𝑥) = 21𝑥 + 37𝑥 + 12 and which represents the function ℎ(𝑥) = 21𝑥 2 − 37𝑥 + 12? Explain your reasoning. I just graphed them both on the calculator and I can see that g(x) is k and h(x) is l.

7. Is it more convenient to complete the square for 𝑥 2 + 𝑏𝑥 + 𝑐 when 𝑏 is odd or when 𝑏 is even? Explain. Completing the square involves cutting the second term in half and then squaring it, so it is more convenient to complete the square if that middle term is even so we don’t get fractions.

8. A student is using completing the square to solve the quadratic function. Is their work correct? Explain.

𝑥 2 + 10𝑥 = 12 𝑥 2 + 10𝑥 + 20 = 12 (𝑥 + 5)2 = 12 𝑥 + 5 = ±√12 𝑥 = −5 ± √12

The work is not correct because they added 20 to one side of the function and not to the other side of the function, so the equation isn’t balanced any more. They should have added 20 to the other side 2 to keep it balanced.

𝑥 + 10𝑥 = 12 𝑥 2 + 10𝑥 + 20 = 12 + 20 (𝑥 + 5)2 = 32 𝑥 + 5 = ±√32 𝑥 = −5 ± √32

Name: ________________________________________________Date: _______________________Hour: ___________ 9. A student is using the quadratic formula to solve the equation −3𝑥 2 + 7𝑥 = 5. Is their work correct? Explain.

−7 ± √72 − 4(−3)(5) 𝑥= 2(−3) −7 ± √109 = −6 𝑥 ≈ −0.57 𝑎𝑛𝑑 𝑥 ≈ 2.91

The work is not correct because the equation was not in standard form when they started. It needs to be 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 before you can start the quadratic formula. To get it in standard form we would need to -5 from both sides. −3𝑥 2 + 7𝑥 = 5 −5 −5 −3𝑥 2 + 7𝑥 − 5 = 0 Now we can do the quadratic formula. a=-3, b=7 and c=-5 (not 5).

10. While standing at the top of a 25 foot ladder you drop a paintbrush and it freefalls to the ground. How many seconds will it take the paintbrush to hit the ground? The gravity pulls on it with -16x2. Since we are just dropping it, there is no middle term. It started 25ft high, so that is the last term. We model the fall with 𝑦 = −16𝑥 2 + 25 Since we are finding when it hits the ground, we are solving when y=0. 0 = −16𝑥 2 + 25 −25 − 25 −25 = −16𝑥 2 ̅̅̅̅̅̅ ̅̅̅̅̅̅ −16 −16 1.5625 = 𝑥 2 √1.5625 = √𝑥 2 1.25 = 𝑥 𝑜𝑟 − 1.25 = 𝑥 Since it is seconds, we want the positive answer. It takes 1.25 seconds for the paintbrush to hit the ground.

11. The Parthenon in Athens, Greece, is an ancient structure that has a rectangular base. The length of the base of the Parthenon is 3 meters more than twice its width. The area of the base is about 54 square meters. Find the length and width of the base. Area=basexwidth, so we know the area (54). We don’t know the width, so let’s call it w. The base is 3 meters more than twice its width. That means that the base=2xwidth+3 or 𝑏 = 2𝑤 + 3. So we have area=basexwidth or 54 = 𝑤(2𝑤 + 3). Now just solve for w. 54 = 𝑤(2𝑤 + 3) Multiply w times everything in the parenthesis. 54 = 2𝑤 2 + 3𝑤 Now if we subtract 54 from both sides then the equation is in standard form and we can solve it however you want (graphing, factoring, completing the square, quadratic formula, or calculator). 0 = 2𝑤 2 + 3𝑤 − 54 The solutions are w=-6 and w=4.5.

12. Your friend is throwing rocks again. They are standing on the edge of a cliff and when the stone leaves their hand it is 12 feet away from the water. They throw it straight up with a velocity of 20 feet per second. How many seconds will it take for the rock to hit the water? This is modeled with 𝑦 = −16𝑥 2 + 20𝑥 + 12 because the gravity is -16x2 and the 20x is the throwing it up in the air and the 12 is because it is 12 feet in the air. Now we want to know when the rock hits the water, so we want to know the roots of the equation or when the equation is equal to 0. 0 = −16𝑥 2 + 20𝑥 + 12

Name: ________________________________________________Date: _______________________Hour: ___________ Solve this using graphing, factoring, completing the square, quadratic formula, or the calculator. The solutions are

or about 1.69 and about -0.4.

13. Two students working on a math problem came up with different answers. Student 1 thinks that since they got different answers one of them must be wrong. Student 2 things they are both right. Using the equations below, justify which student is correct. Student 1: 𝑦 = 𝑥 2 + 6𝑥 − 7 Student 2: 𝑦 = (𝑥 + 3)2 − 16 Student 2 looks like they completed the square student didn’t. If we complete the square on student 1’s equation then maybe it will look like student 2’s. Let’s see: 𝑦 = 𝑥 2 + 6𝑥 − 7 To complete the square we half the middle term and then square it.

6 2

= 3 𝑎𝑛𝑑 32 = 9 so we add that

to each side. 𝑦 + 9 = 𝑥 2 + 6𝑥 + 9 − 7 Now the right side can be written as a quantity squared. 𝑦 + 9 = (𝑥 + 3)2 − 7 Now get the y alone by subtracting 9 from each side. 𝑦 = (𝑥 + 3)2 − 16 They are the same! They are both correct.

14. Brett and Andre each threw a baseball into the air. The height of Brett’s ball is modeled by the quadratic function ℎ(𝑡) = −16𝑡 2 + 79𝑡 + 6, where ℎ is the height in feet and 𝑡 is the time in seconds. The height of Andre’s baseball is given by the graph below. Brett claims that his baseball was in the air longer than Andres. Andre thinks his was in the air for longer. Whose ball took the longer amount of time to hit the ground? We can see in the picture that Andre’s ball hit the ground just after 4 seconds because the graph crosses the x-axis just past 4. If we find the roots of Brett’s (by graphing, factoring, completing the square, quadratic formula, or calculator) we can see that his roots are about at -0.07 and 5.01. The negative root doesn’t make sense in this context (because you can’t have negative seconds) so we know that his took just over 5 seconds to hit the ground. That means that Brett’s ball took a longer amount of time to hit the ground.

15. Use two different methods to find the roots of the quadratic function 𝑓(𝑥) = 𝑥 2 − 1𝑥 − 12. We have learned about 5 methods. These are 5 different ways to solve the same problem, so we should get the same answer every time. Here they are: Graphing: Graph the function 𝑓(𝑥) = 𝑥 2 − 1𝑥 − 12 and have your graph find the zeros by pressing “menu”, “analyze graph”, and “Zero”. Highlight where the graph crosses the x-axis on one side and hit enter. Do the same thing to the other intersection? The two zeros are x=-3 and x=4. Factoring In order to factor we need to find two numbers that multiply to -12 and add to -1. Make a list of all of the numbers that multiply to -12, and then see what they add to: -1 and 12 adds to 11 1 and -12 adds to -11 -2 and 6 adds to 4 2 and -6 adds to -4

Name: ________________________________________________Date: _______________________Hour: ___________ -3 and 4 adds to 1 3 and -4 adds to -1 so the factors must be 3 and -4. That means we could write 𝑓(𝑥) = 𝑥 2 − 1𝑥 − 12 as 𝑓(𝑥) = (𝑥 + 3)(𝑥 − 4). This is easier to solve because we can solve each piece individually. 𝑥 + 3 = 0 Get the x alone by subtracting 3 from each side. −3 −3 𝑥 = −3 is one of the solutions. 𝑥 − 4 = 0 Get the x alone by adding 4 to each side. +4 +4 𝑥 = 4 is the other of the solutions. Completing the Square In order to complete the square, we half the middle term and then square it, and add that to both sides. 0 = 𝑥 2 − 1𝑥 − 12 Half of -1 is ½, and 1

1

12 2

is ¼. So we add ¼ to each side.

0 + = 𝑥 − 1𝑥 + − 12 Now we can write it as the quantity squared. 2

4

1 4

4

1 2

= (𝑥 − ) − 12 Now just solve it by getting the x alone. Add 11 ¾ to each side. 2

1 2

1

12 = (𝑥 − ) Square root both sides. (Don’t forget that square rooting gives two answers, a + and a -.) 4 1

1

2

2

2

3 =𝑥−

1

1

2

2

𝑎𝑛𝑑 − 3 = 𝑥 − Solve each one for x by adding ½ to each side.

4 = 𝑥 𝑎𝑛𝑑 − 3 = 𝑥 Those are our two solutions. Quadratic Formula The equation 𝑓(𝑥) = 𝑥 2 − 1𝑥 − 12 is in standard form, so we know a=1, b=-1, and c=-12. The quadratic formula is 𝑥 = 𝑥= 𝑥= 𝑥= 𝑥= 𝑥= 𝑥=

−(−1)±√(−1)2 −4(1)(−12) 2(1) 1±√1−−48 2 1±√49 2 1±7 2 1+7 2 1−7 2

−𝑏±√𝑏 2 −4𝑎𝑐 2𝑎

so we just plug in our a, b, and c.

And start simplifying. –(-1) is just 1, and (-1)2 is 1, and 4(1)(-12) is -48 and 2(1) is 2.

1- -48 is 49

and the square root of 49 is 7.

Now we can break it down into the + version and the – version. 8

= = 4 so x=4 is one solution. =

2 −6 2

= −3 so x=-3 is the other solution.

Calculator You can solve this on your calculator by opening a calculator page and then pressing “menu”, “algebra”, “polynomial tools”, the “find roots of polynomial”. This is a degree of 2 so we hit “ok”. The first term is 1, then -1, then -12, so we type it in like the screen below, then hit okay.

Hit enter and it tells you that the roots are -3 and 4, so x=-3 and x=4.