# High School: Algebra

### Reasoning with Equations and Inequalities HSA-REI.C.7

7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.

Students should know that this is a slight expansion on the previous standard. Our only change is that we instead of having two lines, we have a linear equation (a straight line) and a quadratic equation (a not so straight line). No big deal.

In the simplest terms, the quadratic equation is just a linear equation with a square sign over a variable. It can have more terms, but as long as the largest exponent over a variable is 2, it's a quadratic equation. Simple enough, right?

Students must know the quadratic formula:

Actually, they should have it tattooed on their faces. It'll be a great icebreaker at parties.

The variables in the quadratic equation (lovingly named a, b, and c) are derived from our standard quadratic equation ax2 + bx + c = 0.

Let's use the actual problem in the standard itself as an example. We have y = -3x as our linear equation, and x2 + y2 = 3 as our quadratic. (Technically, the second equation isn't a "true" quadratic, but there's no shame in torturing your students a little bit to teach them some more. We're only kidding about the torturing part.)

If we graph these bad boys, we can already see that we have two points of intersection.

We can prove why this is the case algebraically and calculate the exact points of intersection. First, we can solve for our quadratic as we did with linear equations: by substituting one equation into another.

That turns x2 + y2 = 3 into x2 + (-3x)2 = 3. We can eventually get it into the ax2 + bx + c = 0 format (which ends up being 10x2 – 3 = 0) so that we can use the quadratic formula. In our case, a = 10, b = 0, and c = -3.

If we plug in those values into the quadratic formula, we get

x ≈ ±0.55

Since we know what x is now, we can simply plug those values into our y = -3x equation to get our answers of ±1.64. That means our points of intersection are (-0.55, 1.64) and (0.55, -1.64). It matches up with the graph, so we're done.

Students should also make use of the discriminant in order to figure out whether the line and quadratic intersect once, twice, or not at all. The discriminant is the part of the quadratic formula under the radical: b2 – 4ac.

If the value of the discriminant is less than 0, there are no points of intersection. If the answer is equal to 0, there is one point of intersection (a tangent line). If the answer is greater than 0, there are two points of intersection.

The value of our discriminant is (0)2 – (4 × 10 × -3) = 120. That means there are two points of intersection. Using the graph, the quadratic formula, and this little tidbit, we've triple checked our answer. Talk about thorough.

#### Drills

1. Your evil math teacher (we're only half kidding) has decided to make your life really difficult by giving you an extra hard problem to solve. Your teacher has given you y = -10x + 20 and 3x2 + 2y2 = 3, for which you must find the points of intersection but you have to do it without a graph.

No points of intersection

Did your evil math teacher make you do more work than you had to? Before we dive headfirst into calculating the points, we can use the discriminant to figure out if there are any points of intersection to begin with. Substituting our y value, we get 3x2 + 2(-10x + 20)2 = 3, which simplifies to 203x2 – 800x + 797 = 0. With our newborn a, b, and c values (aren't they adorable?), we can find the discriminant by calculating b2 – 4ac, which will give us -1,287,164. It's a negative number, so there are no points of intersection. Our graph below proves this as well:

2. A traffic report generated by the city council has determined that traffic increases or decreases (depending on the time of day) based on the formula x2 – 10x – 5 = y. Meanwhile the number of traffic tickets given at time x is y = 2x + 3. The value of y when both graphs intersect equals the number of tickets given out. What is the largest of these values?

28 tickets

As usual, we should begin by substituting y = 2x + 3 into x2 – 10x – 5 = y. After arranging it into ax2 + bx + c = 0 form, we get x2 – 12x – 8 = 0. After that, it's quadratic formula time: . Once your calculator stops huffing and puffing, it'll give you x = 12.6 and x = -0.63 as the two intersection points. However, we're looking for the tickets given out (y, not x). Plugging the two x values back into the linear equation gives y values of 28.2 and 1.74, respectively. Hence the answer is (A).

3. The Yummy Gummy Pie Factory (a delicious but fattening industry) has modeled its daily output to be 10x2 + x – 3 = y. Meanwhile, the consumption of its product is modeled by y = 15x. Find where the positive point of intersection occurs.

(1.58, 23.8)

First, have a Yummy Gummy Pie. They're delicious. Second, we solve by substitution, then by using the quadratic equation to get x values of 1.58 and -0.18. After that, we should go back to the linear function to get y values of 23.8 and -2.7, respectively. But we only care about the positive value.

4. What are the points of intersection of 50x2 – 10 = y and y = 7x + 75?

(1.38, 84.5) and (-1.24, 66.3)

If we set the two equations equal to one another, we end up with 50x2 – 10 = 7x + 75, which is 50x2 – 7x – 85 = 0 in the needed format. The quadratic equation will plop out the values x = 1.38 and x = 1.24. That narrows our choices down to (A) or (C). We can find the y values by plugging our x values back into y = 7x + 75, and they'll give (A) as the right answer.

5. What are the points of intersection of 10x2 – 10 = y and y = 5x?

(-0.78, -3.9) and (1.3, 6.4)

The two intersect when the points equal each other, so we can set the equations equal to each other as well. That gives the quadratic equation of 10x2 – 5x – 10 = 0. Plugging the values of 10, -5, and -10 in for a, b, and c, we get x = -0.78 and x = 1.3 as our answers, but those are only the x coordinates. The y coordinates are 5 times those. We end up with (-0.78, -3.9) and (1.3, 6.4).

6. The math whiz kids working for the racetrack owners have estimated revenues to be modeled by x2 + 10x – 78 = y while the costs are y = 17x (where y is in hundreds of dollars and x is in hundreds of tickets sold). How many tickets must be sold for the racetrack guys just to break even?

1300

Breaking even is when the revenue and the profits are equal. In essence this means when x2 + 10x – 78 = 17x. If we rearrange this and use the quadratic formula to find x, we'll end up with -6 and 13 as our possible values. But that doesn't mean (A) is the answer. The variable x represents hundreds of tickets sold, not individual tickets. That means our answer is actually (C).

7. How much revenue will the racetrack owners from the previous question earn if they only break even?

\$22,100

We already found our x coordinate in the previous problem. All that remains now is plugging in 13 into the linear relationship y = 17x. That gives us y = 221. However, the problem states that y is in hundreds of dollars, so we have to report our findings as such. (If you think about it, \$221 for 1,300 seats means \$0.17 per ticket, which is highly unlikely unless we're in the 1920's or something.)

8. In order to solve a problem where we are asked to find a point of intersection between a linear equation an equation such as this: 166x2 + 3x – 13 = y, we must utilize what to help us?

The quadratic equation is what's given in the question. The quadratic formula, on the other hand, is . That will give you the x values without wanting to pull your hair out or wanting to shave your math teacher's head in their sleep (which we're pretty sure is illegal anyway).

9. Which of the following is a quadratic equation in standard form?

10x2 + x – 3 = 0

All of the equations above can be formed into ax2 + bx + c = 0 form eventually, but as of right now, only (A) is like that! We know, we know, it's a cheap shot, but what can we do?

10. Even if you're pickier than an orangutan cleaning his friend, when looking to find the points of intersection on a graph, you can always come up with which of the following?

An approximation