document.write( "Question 389645: Solving Quadratic Equations by factoring (when first cioefficient is negative).\r
\n" ); document.write( "\n" ); document.write( "pls walk me thru each step: \r
\n" ); document.write( "\n" ); document.write( "-5x^2 + 2x +3 = 0\r
\n" ); document.write( "\n" ); document.write( "i know we multiply by -1 to change each sign:
\n" ); document.write( "5x^2 -2x -3. then we factor. but I am being told that the factors are\r
\n" ); document.write( "\n" ); document.write( "(5x + 3) (x - 1) = 0. I do not understand this. Why isn't it -3, & +1???\r
\n" ); document.write( "\n" ); document.write( "I just don't understand the answer to be x = - 3/5 or x = 1, Seems it should be the opposite.\r
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\n" ); document.write( "\n" ); document.write( "thank you \n" ); document.write( "

Algebra.Com's Answer #276160 by MathLover1(20850) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( " $\"-5x%5E2+%2B+2x+%2B3+=+0\"$ ........ multiply by $\"-1+\"$ to change each sign\r
\n" ); document.write( "\n" ); document.write( " $\"5x%5E2+-2x+-3\"$ \r
\n" ); document.write( "\n" ); document.write( " then we factor\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

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

Now multiply the first coefficient $\"5\"$ by the last term $\"-3\"$ to get $\"%285%29%28-3%29=-15\"$ .

Now the question is: what two whole numbers multiply to $\"-15\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?

To find these two numbers, we need to list all of the factors of $\"-15\"$ (the previous product).

Factors of $\"-15\"$ :

1,3,5,15

-1,-3,-5,-15

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

These factors pair up and multiply to $\"-15\"$ .

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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-2\"$ :

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First Number	Second Number	Sum
1	-15	1+(-15)=-14
3	-5	3+(-5)=-2
-1	15	-1+15=14
-3	5	-3+5=2

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

So the two numbers $\"3\"$ and $\"-5\"$ both multiply to $\"-15\"$ and add to $\"-2\"$

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

$\"5x%5E2%2Bhighlight%283x-5x%29-3\"$ Replace the second term $\"-2x\"$ with $\"3x-5x\"$ .

$\"%285x%5E2%2B3x%29%2B%28-5x-3%29\"$ Group the terms into two pairs.

$\"x%285x%2B3%29%2B%28-5x-3%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%285x%2B3%29-1%285x%2B3%29\"$ Factor out $\"1\"$ 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.

$\"%28x-1%29%285x%2B3%29\"$ Combine like terms. Or factor out the common term $\"5x%2B3\"$

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Answer:

So $\"5%2Ax%5E2-2%2Ax-3\"$ factors to $\"%28x-1%29%285x%2B3%29\"$ .

In other words, $\"5%2Ax%5E2-2%2Ax-3=%28x-1%29%285x%2B3%29\"$ .

Note: you can check the answer by expanding $\"%28x-1%29%285x%2B3%29\"$ to get $\"5%2Ax%5E2-2%2Ax-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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\n" ); document.write( "\n" ); document.write( " $\"%285x%2B3%29%28x-1%29\"$ ...set to zero\r
\n" ); document.write( "\n" ); document.write( " $\"%285x%2B3%29%28x-1%29=0\"$ ........product is equal to zero if at least one or both factors are equal to zero, so you will have\r
\n" ); document.write( "\n" ); document.write( " $\"5x%2B3=0\"$ .....-> $\"5x=-3\"$ .....-> $\"x=-3%2F5\"$ ....or\r
\n" ); document.write( "\n" ); document.write( " $\"x-1=0\"$ ......-> $\"x=+1\"$ ...\r
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\n" ); document.write( "\n" ); document.write( "the answer is: $\"x+=+-3%2F5\"$ or $\"x+=+1\"$ \r
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\n" ); document.write( "\n" ); document.write( "another way to check the result:\r
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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 $\"5%2Ax%5E2-2%2Ax-3=0\"$ ( notice $\"a=5\"$ , $\"b=-2\"$ , and $\"c=-3\"$ )
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\n" ); document.write( " $\"x+=+%28--2+%2B-+sqrt%28+%28-2%29%5E2-4%2A5%2A-3+%29%29%2F%282%2A5%29\"$ Plug in a=5, b=-2, and c=-3
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\n" ); document.write( " $\"x+=+%282+%2B-+sqrt%28+%28-2%29%5E2-4%2A5%2A-3+%29%29%2F%282%2A5%29\"$ Negate -2 to get 2
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\n" ); document.write( " $\"x+=+%282+%2B-+sqrt%28+4-4%2A5%2A-3+%29%29%2F%282%2A5%29\"$ Square -2 to get 4 (note: remember when you square -2, you must square the negative as well. This is because $\"%28-2%29%5E2=-2%2A-2=4\"$ .)
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\n" ); document.write( " $\"x+=+%282+%2B-+sqrt%28+4%2B60+%29%29%2F%282%2A5%29\"$ Multiply $\"-4%2A-3%2A5\"$ to get $\"60\"$
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\n" ); document.write( " $\"x+=+%282+%2B-+sqrt%28+64+%29%29%2F%282%2A5%29\"$ Combine like terms in the radicand (everything under the square root)
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\n" ); document.write( " $\"x+=+%282+%2B-+8%29%2F%282%2A5%29\"$ 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( " $\"x+=+%282+%2B-+8%29%2F10\"$ Multiply 2 and 5 to get 10
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\n" ); document.write( " So now the expression breaks down into two parts
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\n" ); document.write( "
\n" ); document.write( " $\"x+=+%282+%2B+8%29%2F10\"$ or $\"x+=+%282+-+8%29%2F10\"$
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\n" ); document.write( "
\n" ); document.write( " Lets look at the first part:
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\n" ); document.write( " $\"x=%282+%2B+8%29%2F10\"$
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\n" ); document.write( " $\"x=10%2F10\"$ Add the terms in the numerator
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\n" ); document.write( " $\"x=1\"$ Divide
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\n" ); document.write( " So one answer is
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\n" ); document.write( " $\"x=1\"$
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\n" ); document.write( " Now lets look at the second part:
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\n" ); document.write( " $\"x=%282+-+8%29%2F10\"$
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\n" ); document.write( " $\"x=-6%2F10\"$ Subtract the terms in the numerator
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\n" ); document.write( " $\"x=-3%2F5\"$ Divide
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\n" ); document.write( " So another answer is
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\n" ); document.write( " $\"x=-3%2F5\"$
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\n" ); document.write( " So our solutions are:
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\n" ); document.write( " $\"x=1\"$ or $\"x=-3%2F5\"$
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