SOLUTION: Verify that -2 is a root of the equation {{{2x^3+x^2-10x-8=0}}} Find the other two roots, correct to 2 decimal places.

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Question 263378: Verify that -2 is a root of the equation 2x%5E3%2Bx%5E2-10x-8=0
Find the other two roots, correct to 2 decimal places.
Found 2 solutions by drk, richwmiller:
Answer by drk(1908) About Me  (Show Source):
You can put this solution on YOUR website!
we can use synthetic division to verify. This will generate another equation that we will factor.
-2 // . . . 2 . . . . 1 . . . . -10 . . . . . -8
. . . . . . . . . . . . -4 . . . . .6 . . . . . . .8
. . . . . . . 2 . . . . -3 . . . . -4 . . . // . 0
So, we have verified by getting a 0 remainder. We now have a new equation of the form
2x%5E2+-+3x+-+4+=+0
By quadratic, we get
x+=+%283+%2B-+sqrt%289-4%2A2%2A%28-4%29%29%29%2F%284%29
and then
x+=+%283+%2B-+sqrt%2841%29%29%2F%284%29
we have two answers and rounded, we get
x ~ 2.35
x ~ -0.85

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
(x+2)(2x^2-3 x-4) = 0
there are several ways of veryfying that -2 is a solution
You can plug -2 in for x and see if the equation comes out equal
You can factor which I did.
and here are the other two solutions
by factoring
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 2x%5E2-3x-4, we can see that the first coefficient is 2, the second coefficient is -3, and the last term is -4.



Now multiply the first coefficient 2 by the last term -4 to get %282%29%28-4%29=-8.



Now the question is: what two whole numbers multiply to -8 (the previous product) and add to the second coefficient -3?



To find these two numbers, we need to list all of the factors of -8 (the previous product).



Factors of -8:

1,2,4,8

-1,-2,-4,-8



Note: list the negative of each factor. This will allow us to find all possible combinations.



These factors pair up and multiply to -8.

1*(-8) = -8
2*(-4) = -8
(-1)*(8) = -8
(-2)*(4) = -8


Now let's add up each pair of factors to see if one pair adds to the middle coefficient -3:



First NumberSecond NumberSum
1-81+(-8)=-7
2-42+(-4)=-2
-18-1+8=7
-24-2+4=2




From the table, we can see that there are no pairs of numbers which add to -3. So 2x%5E2-3x-4 cannot be factored.



===============================================================





Answer:



So 2%2Ax%5E2-3%2Ax-4 doesn't factor at all (over the rational numbers).



So 2%2Ax%5E2-3%2Ax-4 is prime.


and using the quadratic formula
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 2x%5E2%2B-3x%2B-4+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-3%29%5E2-4%2A2%2A-4=41.

Discriminant d=41 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+41+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+41+%29%29%2F2%5C2+=+2.35078105935821
x%5B2%5D+=+%28-%28-3%29-sqrt%28+41+%29%29%2F2%5C2+=+-0.850781059358212

Quadratic expression 2x%5E2%2B-3x%2B-4 can be factored:
2x%5E2%2B-3x%2B-4+=+2%28x-2.35078105935821%29%2A%28x--0.850781059358212%29
Again, the answer is: 2.35078105935821, -0.850781059358212. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-3%2Ax%2B-4+%29