Question

Need Help Please! Not just the answer! Find the quadratic equation with roots -1+4i and -1-4i

Answer 1

If the roots are -1+4i or -1-4i

then x = -1+4i or x = -1-4i

Answer 2

the goal is to get everything in each equation to the left side. This is so you are left with 0 on the right side

Answer 3

so 0=-x-1-4i?

Answer 4

x = -1 + 4i

x+1 = -1 + 4i + 1 ... Add x to both sides.

x+1 = 4i

Answer 5

recall that i = sqrt(-1)

to get rid of that 'i', you square both sides (to undo the square root)

Answer 6

how do you sqrt x+1?

Answer 7

so let's do so...


x+1 = 4i

(x+1)^2 = (4i)^2

(x+1)^2 = 4^2*i^2

(x+1)^2 = 16i^2

(x+1)^2 = 16(-1)

(x+1)^2 = -16

now we can move the last bit to the other side

(x+1)^2+16 = -16+16 ... Add 16 to both sides.

(x+1)^2 + 16 = 0

Answer 8

you square (x+1)

NOT square root (x+1)

Answer 9

Oh! Okay I got you.

Answer 10

so you see how I got (x+1)^2 + 16 = 0 ?

Answer 11

Yes.

Answer 12

ok let's do the same for x = -1 - 4i

Answer 13

you'll see that the steps are nearly identical

Answer 14

Okay, and then do you do the quadratic formula ?

Answer 15

x = -1-4i

x+1 = -4i

(x+1)^2 = (-4i)^2

(x+1)^2 = 16i^2

(x+1)^2 = 16(-1)

(x+1)^2 = -16

(x+1)^2 + 16 = 0

Answer 16

so as you can see, the two roots lead to (x+1)^2 + 16 = 0

Answer 17

you use the quadratic formula to find the roots if you know the quadratic

Answer 18

In this case, we don't know the quadratic but we know the roots

Answer 19

anyways, the last step is to expand and simplify (x+1)^2 + 16

Answer 20

So x^2 + 2x +17?

Answer 21

you nailed it

Answer 22

now if you used the quadratic formula for x^2 + 2x +17, you'll find that the roots are -1+4i and -1-4i

Answer 23

so this is like thinking backwards

Answer 24

Okay I get it now . Thank you so much for your help

Answer 25

you're welcome