document.write( "Question 149608: What three techniques can be used to solve quadratic equations? Demonstrate these techniques on the equation 12x^2-10x-42=0. \n" ); document.write( "

Algebra.Com's Answer #109749 by jim_thompson5910(35256) $\"\"$ $\"About$

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Technique #1 Factoring:\r
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\n" ); document.write( "\n" ); document.write( "First let's factor $\"12x%5E2-10x-42\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2-10x-42\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%286x%5E2-5x-21%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"6x%5E2-5x-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"6x%5E2-5x-21\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-21\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"-21\"$ to get $\"%286%29%28-21%29=-126\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-126\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-126\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-126\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,9,14,18,21,42,63,126\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-126\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-126)
\n" ); document.write( "2*(-63)
\n" ); document.write( "3*(-42)
\n" ); document.write( "6*(-21)
\n" ); document.write( "7*(-18)
\n" ); document.write( "9*(-14)
\n" ); document.write( "(-1)*(126)
\n" ); document.write( "(-2)*(63)
\n" ); document.write( "(-3)*(42)
\n" ); document.write( "(-6)*(21)
\n" ); document.write( "(-7)*(18)
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-5\"$ :\r
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First Number	Second Number	Sum
1	-126	1+(-126)=-125
2	-63	2+(-63)=-61
3	-42	3+(-42)=-39
6	-21	6+(-21)=-15
7	-18	7+(-18)=-11
9	-14	9+(-14)=-5
-1	126	-1+126=125
-2	63	-2+63=61
-3	42	-3+42=39
-6	21	-6+21=15
-7	18	-7+18=11
-9	14	-9+14=5

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"9\"$ and $\"-14\"$ add to $\"-5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"9\"$ and $\"-14\"$ both multiply to $\"-126\"$ and add to $\"-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-5x\"$ with $\"9x-14x\"$ . Remember, $\"9\"$ and $\"-14\"$ add to $\"-5\"$ . So this shows us that $\"9x-14x=-5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2Bhighlight%289x-14x%29-21\"$ Replace the second term $\"-5x\"$ with $\"9x-14x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%286x%5E2%2B9x%29%2B%28-14x-21%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x%2B3%29%2B%28-14x-21%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x%2B3%29-7%282x%2B3%29\"$ Factor out $\"7\"$ 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.\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x-7%29%282x%2B3%29\"$ Combine like terms. Or factor out the common term $\"2x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2-10x-42\"$ factors to $\"2%283x-7%29%282x%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"2%283x-7%29%282x%2B3%29=0\"$ Set the factored expression equal to zero\r
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\n" ); document.write( "\n" ); document.write( "Now set each factor equal to zero:\r
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\n" ); document.write( "\n" ); document.write( " $\"3x-7=0\"$ or $\"2x%2B3=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x=7%2F3\"$ or $\"x=-3%2F2\"$ Now solve for x in each case\r
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\n" ); document.write( "\n" ); document.write( "So our answers are \r
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\n" ); document.write( "\n" ); document.write( " $\"x=7%2F3\"$ or $\"x=-3%2F2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Technique #2 Quadratic Formula:\r
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E2-10x-42=0\"$ Start with the given equation.\r
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\n" ); document.write( "\n" ); document.write( "Let's use the quadratic formula to solve for x\r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29\"$ Start with the quadratic formula\r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%2812%29%28-42%29+%29%29%2F%282%2812%29%29\"$ Plug in $\"a=12\"$ , $\"b=-10\"$ , and $\"c=-42\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%2812%29%28-42%29+%29%29%2F%282%2812%29%29\"$ Negate $\"-10\"$ to get $\"10\"$ . \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+100-4%2812%29%28-42%29+%29%29%2F%282%2812%29%29\"$ Square $\"-10\"$ to get $\"100\"$ . \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+100--2016+%29%29%2F%282%2812%29%29\"$ Multiply $\"4%2812%29%28-42%29\"$ to get $\"-2016\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+100%2B2016+%29%29%2F%282%2812%29%29\"$ Rewrite $\"sqrt%28100--2016%29\"$ as $\"sqrt%28100%2B2016%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+2116+%29%29%2F%282%2812%29%29\"$ Add $\"100\"$ to $\"2016\"$ to get $\"2116\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+sqrt%28+2116+%29%29%2F%2824%29\"$ Multiply $\"2\"$ and $\"12\"$ to get $\"24\"$ . \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B-+46%29%2F%2824%29\"$ Take the square root of $\"2116\"$ to get $\"46\"$ . \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2810+%2B+46%29%2F%2824%29\"$ or $\"x+=+%2810+-+46%29%2F%2824%29\"$ Break up the expression. \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+%2856%29%2F%2824%29\"$ or $\"x+=++%28-36%29%2F%2824%29\"$ Combine like terms. \r
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\n" ); document.write( "\n" ); document.write( " $\"x+=+7%2F3\"$ or $\"x+=+-3%2F2\"$ Simplify. \r
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\n" ); document.write( "\n" ); document.write( "So our answers are $\"x+=+7%2F3\"$ or $\"x+=+-3%2F2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Technique # 3 Completing the square\r
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\n" ); document.write( "\n" ); document.write( " $\"12+x%5E2%2B10+x-42\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"12%28x%5E2%2B%285%2F6%29x-7%2F2%29\"$ Factor out the leading coefficient $\"12\"$ \r
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\n" ); document.write( "\n" ); document.write( "Take half of the x coefficient $\"5%2F6\"$ to get $\"5%2F12\"$ (ie $\"%281%2F2%29%285%2F6%29=5%2F12\"$ ).\r
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\n" ); document.write( "\n" ); document.write( "Now square $\"5%2F12\"$ to get $\"25%2F144\"$ (ie $\"%285%2F12%29%5E2=%285%2F12%29%285%2F12%29=25%2F144\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"12%28x%5E2%2B%285%2F6%29x%2B25%2F144-25%2F144-7%2F2%29\"$ Now add and subtract this value inside the parenthesis. Notice how $\"25%2F144-25%2F144=0\"$ . Since we're adding 0, we're not changing the equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"12%28%28x%2B5%2F12%29%5E2-25%2F144-7%2F2%29\"$ Now factor $\"x%5E2%2B%285%2F6%29x%2B25%2F144\"$ to get $\"%28x%2B5%2F12%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"12%28%28x%2B5%2F12%29%5E2-529%2F144%29\"$ Combine like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"12%28x%2B5%2F12%29%5E2%2B12%28-529%2F144%29\"$ Distribute\r
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\n" ); document.write( "\n" ); document.write( " $\"12%28x%2B5%2F12%29%5E2-529%2F12\"$ Multiply\r
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\n" ); document.write( "\n" ); document.write( "So after completing the square, $\"12x%5E2%2B10x-42\"$ becomes $\"12%28x%2B5%2F12%29%5E2-529%2F12\"$ .\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E2%2B10x-42=0\"$ is equivalent to $\"12%28x%2B5%2F12%29%5E2-529%2F12=0\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"12%28x%2B5%2F12%29%5E2-529%2F12=0\"$ Start with completed square equation.\r
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\n" ); document.write( "\n" ); document.write( " $\"12%28x%2B5%2F12%29%5E2=529%2F12\"$ Add $\"529%2F12\"$ to both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%2B5%2F12%29%5E2=529%2F144\"$ Divide both sides by 12.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%2B5%2F12=0%2B-sqrt%28529%2F144%29\"$ Take the square root of both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%2B5%2F12=sqrt%28529%2F144%29\"$ or $\"x%2B5%2F12=-sqrt%28529%2F144%29\"$ Break up the expression\r
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\n" ); document.write( "\n" ); document.write( " $\"x%2B5%2F12=23%2F12\"$ or $\"x%2B5%2F12=-23%2F12\"$ Take the square root of $\"529%2F144\"$ to get $\"23%2F12\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"x=23%2F12-5%2F12\"$ or $\"x=-23%2F12-5%2F12\"$ Subtract $\"5%2F12\"$ from both sides.\r
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\n" ); document.write( "\n" ); document.write( " $\"x=3%2F2\"$ or $\"x=-7%2F3\"$ Combine like terms and simplify.\r
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\n" ); document.write( "\n" ); document.write( "So the answers are $\"x=3%2F2\"$ or $\"x=-7%2F3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Technique # 4 Graphing\r
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\n" ); document.write( "\n" ); document.write( "Simply graph $\"y=12+x%5E2%2B10+x-42\"$ to get \r
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\n" ); document.write( "\n" ); document.write( " $\"+graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C0%2C+12x%5E2-10x-42%29+\"$ Graph of $\"y=12+x%5E2%2B10+x-42\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now use the calculator's zero function to find the zeros at $\"x=2.333\"$ and $\"x=-1.5\"$ \n" ); document.write( "

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