document.write( "Question 901085: Which is the solution to this equation?\r
\n" ); document.write( "\n" ); document.write( "x^2+3x-10=0\r
\n" ); document.write( "\n" ); document.write( "a.0
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Algebra.Com's Answer #546442 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2%2B3x-10\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-10\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-10\"$ to get $\"%281%29%28-10%29=-10\"$ .

Now the question is: what two whole numbers multiply to $\"-10\"$ (the previous product) and add to the second coefficient $\"3\"$ ?

To find these two numbers, we need to list all of the factors of $\"-10\"$ (the previous product).

Factors of $\"-10\"$ :

1,2,5,10

-1,-2,-5,-10

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

These factors pair up and multiply to $\"-10\"$ .

1*(-10) = -10
2*(-5) = -10
(-1)*(10) = -10
(-2)*(5) = -10

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	-10	1+(-10)=-9
2	-5	2+(-5)=-3
-1	10	-1+10=9
-2	5	-2+5=3

From the table, we can see that the two numbers $\"-2\"$ and $\"5\"$ add to $\"3\"$ (the middle coefficient).

So the two numbers $\"-2\"$ and $\"5\"$ both multiply to $\"-10\"$ and add to $\"3\"$

Now replace the middle term $\"3x\"$ with $\"-2x%2B5x\"$ . Remember, $\"-2\"$ and $\"5\"$ add to $\"3\"$ . So this shows us that $\"-2x%2B5x=3x\"$ .

$\"x%5E2%2Bhighlight%28-2x%2B5x%29-10\"$ Replace the second term $\"3x\"$ with $\"-2x%2B5x\"$ .

$\"%28x%5E2-2x%29%2B%285x-10%29\"$ Group the terms into two pairs.

$\"x%28x-2%29%2B%285x-10%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x-2%29%2B5%28x-2%29\"$ Factor out $\"5\"$ 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.

$\"%28x%2B5%29%28x-2%29\"$ Combine like terms. Or factor out the common term $\"x-2\"$

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Answer:

So $\"x%5E2%2B3%2Ax-10\"$ factors to $\"%28x%2B5%29%28x-2%29\"$ .

In other words, $\"x%5E2%2B3%2Ax-10=%28x%2B5%29%28x-2%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B5%29%28x-2%29\"$ to get $\"x%5E2%2B3%2Ax-10\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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Solved by pluggable solver: Quadratic Formula

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

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\n" ); document.write( " Starting with the general quadratic
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\n" ); document.write( " $\"ax%5E2%2Bbx%2Bc=0\"$
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\n" ); document.write( " the general solution using the quadratic equation is:
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\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29\"$
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\n" ); document.write( " So lets solve $\"x%5E2%2B3%2Ax-10=0\"$ ( notice $\"a=1\"$ , $\"b=3\"$ , and $\"c=-10\"$ )
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\n" ); document.write( " $\"x+=+%28-3+%2B-+sqrt%28+%283%29%5E2-4%2A1%2A-10+%29%29%2F%282%2A1%29\"$ Plug in a=1, b=3, and c=-10
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\n" ); document.write( " $\"x+=+%28-3+%2B-+sqrt%28+9-4%2A1%2A-10+%29%29%2F%282%2A1%29\"$ Square 3 to get 9
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\n" ); document.write( " $\"x+=+%28-3+%2B-+sqrt%28+9%2B40+%29%29%2F%282%2A1%29\"$ Multiply $\"-4%2A-10%2A1\"$ to get $\"40\"$
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\n" ); document.write( " $\"x+=+%28-3+%2B-+sqrt%28+49+%29%29%2F%282%2A1%29\"$ Combine like terms in the radicand (everything under the square root)
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\n" ); document.write( " $\"x+=+%28-3+%2B-+7%29%2F%282%2A1%29\"$ Simplify the square root (note: If you need help with simplifying the square root, check out this solver)
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\n" ); document.write( " $\"x+=+%28-3+%2B-+7%29%2F2\"$ Multiply 2 and 1 to get 2
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\n" ); document.write( " So now the expression breaks down into two parts
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\n" ); document.write( " $\"x+=+%28-3+%2B+7%29%2F2\"$ or $\"x+=+%28-3+-+7%29%2F2\"$
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\n" ); document.write( " Lets look at the first part:
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\n" ); document.write( " $\"x=%28-3+%2B+7%29%2F2\"$
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\n" ); document.write( " $\"x=4%2F2\"$ Add the terms in the numerator
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\n" ); document.write( " $\"x=2\"$ Divide
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\n" ); document.write( " So one answer is
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\n" ); document.write( " $\"x=2\"$
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\n" ); document.write( " Now lets look at the second part:
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\n" ); document.write( " $\"x=%28-3+-+7%29%2F2\"$
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\n" ); document.write( " $\"x=-10%2F2\"$ Subtract the terms in the numerator
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\n" ); document.write( " $\"x=-5\"$ Divide
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\n" ); document.write( " So another answer is
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\n" ); document.write( " $\"x=-5\"$
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\n" ); document.write( " So our solutions are:
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\n" ); document.write( " $\"x=2\"$ or $\"x=-5\"$
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