Question

What is the equation of the following graph?

Answer 1

Hang on, I need to post it.

Answer 2

There.

Answer 3

Do you know of the roots of a quadratic?

Answer 4

Kind of?

Answer 5

Okay, when you graph a quadratic, the points where the graph intersect the x axis are called the roots. Can you determine the roots of your graph?

Answer 6

4 and 6?

Answer 7

Good. Usually we say the roots are x = 4 and x = 6. Remember how you can factor a quadratic equation down to getting values for x?

Answer 8

Could you refresh my memory?

Answer 9

Sure. To do that we must factor a quadratic now. Take a simple example. \[x^{2} + 4x + 4 = 0\] is factored to \[(x+2)(x+2) = 0\] which means \[x = -2, -2\]

Answer 10

Okay.

Answer 11

See how I went from the equation to the x values? in your problems we will go from the x values back to the original equation.

Answer 12

Yes, I see what you did.

Answer 13

From your graph, we see that \[x = 4, 6\]What is the next step upward using my example?

Answer 14

x^2 + 4 +6?

Answer 15

Not quite, take it step by step. If x = 4,6 This means that \[(x-4)(x-6) = 0\]

Answer 16

Okay.

Answer 17

Understand what I did?

Answer 18

Sort of.

Answer 19

But I do understand some of it.

Answer 20

What we are going to do now is to expand this expression. \[(x-4)(x-6) = 0\]Remember how to do that?

Answer 21

Yeah!

Answer 22

Lets see you do it.

Answer 23

(x^2 - x + 24)

Answer 24

No wait. Hang on. I do this better on paper...

Answer 25

x - 4x + 24

Answer 26

Try this, do you remember distributing?

Answer 27

It's that thing where you spread out the numbers across a parentheses set, isn't it?

Answer 28

Yep. So let's do this. \[x(x-6)-4(x-6)\]You will distribute the x and the -4 across the parentheses.

Answer 29

x^2 and 6x, and -4x and +24?

Answer 30

Yes, yes. Now add everything together taking signs into consideration.

Answer 31

So it'd be x^2 +2x + 24?

Answer 32

Oh, I noticed a mistake, there should be a negative beside the 6x because x multiplies -6

Answer 33

Oh. Yeah. I was wondering about that since how we got to the negative part was confusing me.

Answer 34

Okay. Now the mistake is cleared, what is the new correct expression?

Answer 35

x^2 - 2x + 24?

Answer 36

Not quite. \[x(x-6)-4(x-6) = x^{2}-6x \space\and\ -4x+24\]

Answer 37

Oh.

Answer 38

Now add them together, gives you \[x^{2} -10x+24=0\]

Answer 39

So that's the answer?

Answer 40

That's the equation of your graph.

Answer 41

Thank you.

Answer 42

No problems.