document.write( "Question 263310: The table shows the relationship between x and y in a quadratic equation of the form y = ax^2 + bx + c, where a, b, and c are integers.What is the value of a?\r
\n" ); document.write( "\n" ); document.write( "x 1 2 3 4 5 6 7 \r
\n" ); document.write( "\n" ); document.write( "y -8 5 24 49 80 117 160 \n" ); document.write( "

Algebra.Com's Answer #194037 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
x=1 y=-8
\n" ); document.write( "ax^2+bx+c=y\r
\n" ); document.write( "\n" ); document.write( "c = -a-b-8, x = 1, y = -8
\n" ); document.write( "c = -4 a-2 b+5, x = 2, y = 5
\n" ); document.write( "c = -3 (3 a+b-8), x = 3, y = 24
\n" ); document.write( "c = -16 a-4 b+49, x = 4, y = 49
\n" ); document.write( "c = -a-b-8 and
\n" ); document.write( "c = -4a-2 b+5 and
\n" ); document.write( "c = -3*(3a+b-8) and
\n" ); document.write( "c = -16a-4b+49\r
\n" ); document.write( "\n" ); document.write( "a = 3, b = 4, c = -15
\n" ); document.write( "y=3x^2+4x-15
\n" ); document.write( "y=(x+3)(3x-5)
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"3x%5E2%2B4x-15\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"4\"$ , and the last term is $\"-15\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"-15\"$ to get $\"%283%29%28-15%29=-45\"$ .

Now the question is: what two whole numbers multiply to $\"-45\"$ (the previous product) and add to the second coefficient $\"4\"$ ?

To find these two numbers, we need to list all of the factors of $\"-45\"$ (the previous product).

Factors of $\"-45\"$ :

1,3,5,9,15,45

-1,-3,-5,-9,-15,-45

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

These factors pair up and multiply to $\"-45\"$ .

1*(-45) = -45
3*(-15) = -45
5*(-9) = -45
(-1)*(45) = -45
(-3)*(15) = -45
(-5)*(9) = -45

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	-45	1+(-45)=-44
3	-15	3+(-15)=-12
5	-9	5+(-9)=-4
-1	45	-1+45=44
-3	15	-3+15=12
-5	9	-5+9=4

From the table, we can see that the two numbers $\"-5\"$ and $\"9\"$ add to $\"4\"$ (the middle coefficient).

So the two numbers $\"-5\"$ and $\"9\"$ both multiply to $\"-45\"$ and add to $\"4\"$

Now replace the middle term $\"4x\"$ with $\"-5x%2B9x\"$ . Remember, $\"-5\"$ and $\"9\"$ add to $\"4\"$ . So this shows us that $\"-5x%2B9x=4x\"$ .

$\"3x%5E2%2Bhighlight%28-5x%2B9x%29-15\"$ Replace the second term $\"4x\"$ with $\"-5x%2B9x\"$ .

$\"%283x%5E2-5x%29%2B%289x-15%29\"$ Group the terms into two pairs.

$\"x%283x-5%29%2B%289x-15%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%283x-5%29%2B3%283x-5%29\"$ 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.

$\"%28x%2B3%29%283x-5%29\"$ Combine like terms. Or factor out the common term $\"3x-5\"$

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Answer:

So $\"3%2Ax%5E2%2B4%2Ax-15\"$ factors to $\"%28x%2B3%29%283x-5%29\"$ .

In other words, $\"3%2Ax%5E2%2B4%2Ax-15=%28x%2B3%29%283x-5%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B3%29%283x-5%29\"$ to get $\"3%2Ax%5E2%2B4%2Ax-15\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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Solved by pluggable solver: SOLVE quadratic equation with variable

Quadratic equation $\"ax%5E2%2Bbx%2Bc=0\"$ (in our case $\"3x%5E2%2B4x%2B-15+=+0\"$ ) has the following solutons:
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\n" ); document.write( " $\"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"$
\n" ); document.write( "
\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 $\"b%5E2-4ac\"$ : $\"b%5E2-4ac=%284%29%5E2-4%2A3%2A-15=196\"$ .
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\n" ); document.write( " Discriminant d=196 is greater than zero. That means that there are two solutions: $\"+x%5B12%5D+=+%28-4%2B-sqrt%28+196+%29%29%2F2%5Ca\"$ .
\n" ); document.write( "
\n" ); document.write( " $\"x%5B1%5D+=+%28-%284%29%2Bsqrt%28+196+%29%29%2F2%5C3+=+1.66666666666667\"$
\n" ); document.write( " $\"x%5B2%5D+=+%28-%284%29-sqrt%28+196+%29%29%2F2%5C3+=+-3\"$
\n" ); document.write( "
\n" ); document.write( " Quadratic expression $\"3x%5E2%2B4x%2B-15\"$ can be factored:
\n" ); document.write( " $\"3x%5E2%2B4x%2B-15+=+3%28x-1.66666666666667%29%2A%28x--3%29\"$
\n" ); document.write( " Again, the answer is: 1.66666666666667, -3.\n" ); document.write( "Here's your graph:
\n" ); document.write( " $\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B4%2Ax%2B-15+%29\"$

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