Question

Find the roots of the equation: y=4(x+4)^2-12
Enter the solution in ascending order and round to the nearest hundredth

Answer 1

Step 1: Expand terms.
Step 2: Factor if possible.
Step 3: Use the Quadratic Formula to find the roots

Answer 2

I dont even know what that means.

Answer 3

Multiply out the terms first, then see if you can factor out a coefficient. Then use the quadratic formula, which is
\[\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]to find the roots

Answer 4

How do you multiply the terms out?

Answer 5

\[4(x+4)^2 - 12 = 4(x+4)(x+4) - 12 = 4(x^2+8x+16) - 12\]
\[= 4x^2 +32x + 64 - 12 = 4x^2 +32x + 52\]

Answer 6

\[= 4(x^2+8x+13)\]

Answer 7

and then what ?

Answer 8

Now use the quadratic formula to find the roots. When you have a quadratic equation of the form ax^2+bx+c, you can plug in the coefficients a,b, and c into the formula I posted above and those are your roots.

Answer 9

But what do you do with the 4 in the front of the formula?

Answer 10

You can ignore it and here is why: when you are asked to find the "roots," what this literally means is, "find the x values that make y=0"

If y=0, then here is what you have:

\[0=4(x^2+8x+13)\]
Divide both sides of this equation by 4 and you now have:

\[0=x^2+8x+13\]

Answer 11

then you do quadritic formula and your done?

Answer 12

Yes, the quadratic formula is used to find the roots. Roots are x-intercepts.

Answer 13

I dont know what to do after. Like i got \[-8\pm \sqrt{12di}\div2\]

Answer 14

excuse the di under the radical

Answer 15

Ok so you need to simplify the square root. Factor 12 into it's prime factors. Then use the property of \[\sqrt{a}\sqrt{b}=\sqrt{ab}\]
Example: \[\sqrt{8} = \sqrt{2} \sqrt{4} = 2\sqrt{2}\]

Answer 16

The goal is to be able to simplify the expression as much as possible, so you want to factor out as many coeffiecients as you can and cancel them out. So you're trying to factor the 12 in such a way that you can take the square root of one of it's factors.

Answer 17

okay, than what?

Answer 18

So now you should have
\[\frac{-8\pm2\sqrt{3}}{2}\]
And now you can factor out a 2 in the numerator and cancel with the 2 in the denominator so this leaves you with
\[-4\pm \sqrt{3}\]

Answer 19

The question asks you to round to the nearest hundredth, so at this point you can finally convert into decimal form using a calculator. My calculator gives me 1.73 for the sqrt of 3. So just calculate 4.00 - 1.73

Answer 20

Oops I mean \[4\pm1.73\]

Answer 21

Okay. I dont get how to find the roots stillllllllll

Answer 22

At which point does it start to get confusing?

Answer 23

Like I am so confused with everything. I know how to do the quadratic formula and everything. I'm just atad confused.

Answer 24

Do you understand what "root" means?

Answer 25

Yes

Answer 26

Ok. Whenever you are faced with finding the roots, the first thing you must do is get the equation into the form of
\[Ax^2 + Bx + C \]
That is why I said you needed to expand the terms. Expanding simply means multiplying the terms out to get it into another form--the correct form that you need so that you can then use the Quadratic Formula. The Quadratic Formula tells you what your roots are.

You can expect to have 2 roots for this equation because this biggest term is x^2. If it were x^3 then you would have 3 roots. If it were x^4 then you would have 4 roots. Etc.

So in this case you have 2 roots. The plus/minus sign means that for the first root you need to add the terms; for the second root you need to subtract the two terms. The two terms I am referring to in this case are the 4 and the 1.73 that you get from the Quadratic Formula. I hope that clears it up.