Solve the linear quadratic system by graphing

Question

Y = -2x² + 12x - 17

2× = y - 5

Vertex: Vertex form:

Solutions:

Answer 1

Our outline is the following steps, which you can use for similar problems in the future:

1. Graph the first equation.
2. Graph the second equation.
3. Find the intersection points to get the solution pairs that satisfy both equations.

1 --------------------------------------------------------------------------------------------------------------------------------------

The first equation is a quadratic equation. To graph these, it helps to convert the equation into vertex form, which looks like

y = a(x-h)² + k where h and k are coordinates of the vertex (h, k), and a is a real number.

This is done by using the method completing the square. This step requires us to know how to complete the square, so if you do not recall how to do that, I suggest looking back in your notes. Upon completing the square, you should get

y = -2(x+3)² + 1.

(fill in the intermediate algebra steps.)

To check that this is correct, we look to see if it matches the general vertex form I wrote earlier, and it does: a=-2, h=-3, k=1.

To obtain our vertex from an equation written in vertex form, we identify the coordinates h and k and take the point (h, k) as our vertex. For example, the parabola y = (x-5)^2 - 3 has h=5 and k=-3, so its vertex is (5, -3).

To discover other crucial points on the graph of this quadratic, plug in x values to the left and right of the vertex you identified.

2 --------------------------------------------------------------------------------------------------------------------------------------

I'm not sure whether you meant 2x or 2^x, but I will assume 2x. The second equation is a linear equation. To graph these, isolate y.

This should look familiar to graph now.

3 --------------------------------------------------------------------------------------------------------------------------------------

Inspect the graphs of your two equations now; the points of intersection are the solution pairs.

Step 1 is in my opinion the hardest, so I encourage you ask your teacher how to approach that one if you are still having trouble after attempting it on your own. Best of luck! :)