Question

find all x-intercepts of the graph of
f(x)=(x^2+4x)^2+2(x^2+4x)-3

Answer 1

Firstly, expand / distribute those brackets!

Answer 2

x^4+16x^2+2x^2+8x-3?

Answer 3

x^4+18x^2+8x-3

Answer 4

Almost! You're missing \(8x^3\), which comes from the square at the beginning of the expression. The x-intercepts are where f(x)=0, so \[x^4+8x^3+18x^2+8x-3=0\]You'll need to start with trial and error to find the first root of the equation. Pick a value for x, put it in the equation, and see if it equals 0. (hint, it must be a factor of -3).

Answer 5

(x+1) will be a factor

Answer 6

im still confused.

Answer 7

If \[x^4+8x^3+18x^2+8x-3=0\], then for some value of x (i.e. a root of the equation), this equation wil be true. Let's say you try x=1, you get:
\[1^4+8(1)^3+18(1)^2+8(1)-3=1+8+18+8-3=32\neq0\]So 1 is not a root of the equation. Try another number!

Answer 8

i tryed 2 and 3

Answer 9

Did they work?

Answer 10

no
the number just rises

Answer 11

or total

Answer 12

Alright then, try another number. Remember, you're trying factors of -3 (because that's the number at the end of the equation).
The factors of -3 are: -3, -1, 1, 3. We've tried 1, and 3 already, so now try -3, and -1!

Answer 13

ohhhh

Answer 14

so its -3

Answer 15

That's one of them, yep. Step 2: Divide the corresponding factor into your original expression. If x=-3 is a root, then x+3 is a factor.

________________________
x + 3 |\(x^4+8x^3+18x^2+8x-3\)

Do you know how to to long division of expressions?

Answer 16

yes.

Answer 17

after factorisation, the expression will become:
(x+1)(x+3)(x^2+4x-1)=0
so, the x- intercepts are coming -1,-3 and \[-2\pm \sqrt{5}\]

Answer 18

huh?

Answer 19

How did you get on with the long division?

Answer 20

i got x^3+5x^2+3x-1

Answer 21

Bingo! Now, the next step is to find another root the same way we did the first one. Pick a factor of -1 (because -1 is the last number in the expression), and see if the whole thing equals 0.

Answer 22

yes it does

Answer 23

Which factor of -1 did you pick?

Answer 24

-1

Answer 25

Right, so if the root of the equation is -1, the factor of the equation is x+1. Divide this into the equation:

________________
x + 1 |\(x^3+5x^2+3x-1\)

Answer 26

x^2+4x-1

Answer 27

Correct! Now, you've got an expression in the form \[ax^2+bx+c=0\], so you can normally solve that one by factorising, or by using the quadratic formula (factorising isn't possible here unfortunately).

Answer 28

ok 1 sec im going to try quadratic formula

Answer 29

so its \[(-4\pm \sqrt{20})/2\]

Answer 30

Yep! Can you simplify that any further?

Answer 31

yeah to \[-2\pm \sqrt{20}\]

Answer 32

Oooh, close! Don't forget that the \(\sqrt{20}\) is still being divided by 2.

Answer 33

And that \(\sqrt{20}=\sqrt{5}\sqrt{4}\)

Answer 34

ah so it actually \[-2\pm \sqrt{5}\]

Answer 35

Bingo. So to recap what we've done, the 4 roots of the equation \(x^4+8x^3+18x^2+8x-3=0\) are x=-1
x=-3
x=-2+\(\sqrt{5}\)
x=-2-\(\sqrt{5}\)
The roots are also called the 'zeroes', because they are where f(x) will equal zero (i.e. where the function intercepts the x-axis).