SOLUTION: I am having problems with these problems Solve by using the quadratic formula x^2-x-2=0 and also 4x^2-3x+3=0 Thank you

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Question 92942: I am having problems with these problems
Solve by using the quadratic formula
x^2-x-2=0
and also 4x^2-3x+3=0
Thank you
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
"Solve by using the quadratic formula
x^2-x-2=0"

Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve x%5E2-x-2=0 ( notice a=1, b=-1, and c=-2)

x+=+%28--1+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A-2+%29%29%2F%282%2A1%29 Plug in a=1, b=-1, and c=-2



x+=+%281+%2B-+sqrt%28+%28-1%29%5E2-4%2A1%2A-2+%29%29%2F%282%2A1%29 Negate -1 to get 1



x+=+%281+%2B-+sqrt%28+1-4%2A1%2A-2+%29%29%2F%282%2A1%29 Square -1 to get 1 (note: remember when you square -1, you must square the negative as well. This is because %28-1%29%5E2=-1%2A-1=1.)



x+=+%281+%2B-+sqrt%28+1%2B8+%29%29%2F%282%2A1%29 Multiply -4%2A-2%2A1 to get 8



x+=+%281+%2B-+sqrt%28+9+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



x+=+%281+%2B-+3%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%281+%2B-+3%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

x+=+%281+%2B+3%29%2F2 or x+=+%281+-+3%29%2F2

Lets look at the first part:

x=%281+%2B+3%29%2F2

x=4%2F2 Add the terms in the numerator
x=2 Divide

So one answer is
x=2



Now lets look at the second part:

x=%281+-+3%29%2F2

x=-2%2F2 Subtract the terms in the numerator
x=-1 Divide

So another answer is
x=-1

So our solutions are:
x=2 or x=-1

Notice when we graph x%5E2-x-2, we get:

+graph%28+500%2C+500%2C+-11%2C+12%2C+-11%2C+12%2C1%2Ax%5E2%2B-1%2Ax%2B-2%29+

and we can see that the roots are x=2 and x=-1. This verifies our answer





"Solve by using the quadratic formula
4x^2-3x+3=0 "


Let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve 4%2Ax%5E2-3%2Ax%2B3=0 ( notice a=4, b=-3, and c=3)

x+=+%28--3+%2B-+sqrt%28+%28-3%29%5E2-4%2A4%2A3+%29%29%2F%282%2A4%29 Plug in a=4, b=-3, and c=3



x+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%2A4%2A3+%29%29%2F%282%2A4%29 Negate -3 to get 3



x+=+%283+%2B-+sqrt%28+9-4%2A4%2A3+%29%29%2F%282%2A4%29 Square -3 to get 9 (note: remember when you square -3, you must square the negative as well. This is because %28-3%29%5E2=-3%2A-3=9.)



x+=+%283+%2B-+sqrt%28+9%2B-48+%29%29%2F%282%2A4%29 Multiply -4%2A3%2A4 to get -48



x+=+%283+%2B-+sqrt%28+-39+%29%29%2F%282%2A4%29 Combine like terms in the radicand (everything under the square root)



x+=+%283+%2B-+i%2Asqrt%2839%29%29%2F%282%2A4%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



x+=+%283+%2B-+i%2Asqrt%2839%29%29%2F%288%29 Multiply 2 and 4 to get 8



After simplifying, the quadratic has roots of

x=3%2F8+%2B+sqrt%2839%29%2F8%2Ai or x=3%2F8+-+sqrt%2839%29%2F8%2Ai

Notice if we graph the quadratic y=4%2Ax%5E2-3%2Ax%2B3, we get

+graph%28+500%2C+500%2C+-14.625%2C+15.375%2C+-12.5625%2C+17.4375%2C+4%2Ax%5E2-3%2Ax%2B3%29+ graph of y=4%2Ax%5E2-3%2Ax%2B3

And we can see that there are no real roots

To visually verify the answer, check out this page to see a visual representation of imaginary roots