document.write( "Question 260712: y = -x2 - 4x - 3 of this equation
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Algebra.Com's Answer #192062 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
vertex (-2,1)
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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"-1x%5E2%2B-4x%2B-3+=+0\"$ ) 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 $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%28-4%29%5E2-4%2A-1%2A-3=4\"$ .
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\n" ); document.write( " Discriminant d=4 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28--4%2B-sqrt%28+4+%29%29%2F2%5Ca\"$ .
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\n" ); document.write( " $\"x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+4+%29%29%2F2%5C-1+=+-3\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%28-4%29-sqrt%28+4+%29%29%2F2%5C-1+=+-1\"$
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\n" ); document.write( " Quadratic expression $\"-1x%5E2%2B-4x%2B-3\"$ can be factored:
\n" ); document.write( " $\"-1x%5E2%2B-4x%2B-3+=+-1%28x--3%29%2A%28x--1%29\"$
\n" ); document.write( " Again, the answer is: -3, -1.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+-1%2Ax%5E2%2B-4%2Ax%2B-3+%29\"$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"-x%5E2-4%2Ax-3\"$ Start with the given expression.

$\"-%28x%5E2%2B4x%2B3%29\"$ Factor out the GCF $\"-1\"$ .

Now let's try to factor the inner expression $\"x%5E2%2B4x%2B3\"$

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Looking at the expression $\"x%5E2%2B4x%2B3\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"4\"$ , and the last term is $\"3\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"3\"$ to get $\"%281%29%283%29=3\"$ .

Now the question is: what two whole numbers multiply to $\"3\"$ (the previous product) and add to the second coefficient $\"4\"$ ?

To find these two numbers, we need to list all of the factors of $\"3\"$ (the previous product).

Factors of $\"3\"$ :

1,3

-1,-3

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

These factors pair up and multiply to $\"3\"$ .

1*3 = 3
(-1)*(-3) = 3

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"4\"$ :

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First Number	Second Number	Sum
1	3	1+3=4
-1	-3	-1+(-3)=-4

From the table, we can see that the two numbers $\"1\"$ and $\"3\"$ add to $\"4\"$ (the middle coefficient).

So the two numbers $\"1\"$ and $\"3\"$ both multiply to $\"3\"$ and add to $\"4\"$

Now replace the middle term $\"4x\"$ with $\"x%2B3x\"$ . Remember, $\"1\"$ and $\"3\"$ add to $\"4\"$ . So this shows us that $\"x%2B3x=4x\"$ .

$\"x%5E2%2Bhighlight%28x%2B3x%29%2B3\"$ Replace the second term $\"4x\"$ with $\"x%2B3x\"$ .

$\"%28x%5E2%2Bx%29%2B%283x%2B3%29\"$ Group the terms into two pairs.

$\"x%28x%2B1%29%2B%283x%2B3%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B1%29%2B3%28x%2B1%29\"$ Factor out $\"3\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

$\"%28x%2B3%29%28x%2B1%29\"$ Combine like terms. Or factor out the common term $\"x%2B1\"$

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So $\"-1%28x%5E2%2B4x%2B3%29\"$ then factors further to $\"-%28x%2B3%29%28x%2B1%29\"$

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Answer:

So $\"-x%5E2-4%2Ax-3\"$ completely factors to $\"-%28x%2B3%29%28x%2B1%29\"$ .

In other words, $\"-x%5E2-4%2Ax-3=-%28x%2B3%29%28x%2B1%29\"$ .

Note: you can check the answer by expanding $\"-%28x%2B3%29%28x%2B1%29\"$ to get $\"-x%5E2-4%2Ax-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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