document.write( "Question 263378: Verify that -2 is a root of the equation $\"2x%5E3%2Bx%5E2-10x-8=0\"$
\n" ); document.write( "Find the other two roots, correct to 2 decimal places. \n" ); document.write( "

Algebra.Com's Answer #194090 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
(x+2)(2x^2-3 x-4) = 0
\n" ); document.write( "there are several ways of veryfying that -2 is a solution
\n" ); document.write( "You can plug -2 in for x and see if the equation comes out equal
\n" ); document.write( "You can factor which I did.
\n" ); document.write( "and here are the other two solutions\r
\n" ); document.write( "\n" ); document.write( "by factoring
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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 $\"-3\"$ :

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First Number	Second Number	Sum
1	-8	1+(-8)=-7
2	-4	2+(-4)=-2
-1	8	-1+8=7
-2	4	-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.

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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.

\n" ); document.write( "\n" ); document.write( "and using the quadratic formula
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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:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
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\n" ); document.write( " For these solutions to exist, the discriminant $\"b%5E2-4ac\"$ should not be a negative number.
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\n" ); document.write( " First, we need to compute the discriminant $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%28-3%29%5E2-4%2A2%2A-4=41\"$ .
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\n" ); document.write( " Discriminant d=41 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28--3%2B-sqrt%28+41+%29%29%2F2%5Ca\"$ .
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\n" ); document.write( " $\"x%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+41+%29%29%2F2%5C2+=+2.35078105935821\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%28-3%29-sqrt%28+41+%29%29%2F2%5C2+=+-0.850781059358212\"$
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\n" ); document.write( " Quadratic expression $\"2x%5E2%2B-3x%2B-4\"$ can be factored:
\n" ); document.write( " $\"2x%5E2%2B-3x%2B-4+=+2%28x-2.35078105935821%29%2A%28x--0.850781059358212%29\"$
\n" ); document.write( " Again, the answer is: 2.35078105935821, -0.850781059358212.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+2%2Ax%5E2%2B-3%2Ax%2B-4+%29\"$

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