document.write( "Question 584022: 12x^2-10x-42=0
\n" ); document.write( "I have to solve this quadratic equation two different ways and I need to show my work and explain what I am doing. I have no clue where to even start. I didn't even know there was two ways to solve this.....Any help please? \n" ); document.write( "
Algebra.Com's Answer #372685 by KMST(5328)\"\" \"About 
You can put this solution on YOUR website!
\"12x%5E2-10x-42=0\" is equivalent to \"6x%5E2-5x-21=0\"
\n" ); document.write( "I would say there are 3 ways to solve it, and each person may have one preferred way.
\n" ); document.write( "You could:
\n" ); document.write( "factor the polynomial, or
\n" ); document.write( "use the quadratic formula, or
\n" ); document.write( "\"complete the square.\"
\n" ); document.write( "FACTORING (not easy in this case):
\n" ); document.write( "Factoring is easier when the coefficient of x^2 is 1. It is easiest if the solutions and coefficients are all integers. It will not work if the solutions are not rational numbers.
\n" ); document.write( "We are looking for two factors whose product is \"6x%5E2-5x-21=0\". The products of factors like \"%286x-21%29%28x%2B1%29\" or \"%282x-7%29%283x%2B3%29\" will give you the terms \"6x%5E2\" and \"-21\", but you need to get the term \"-5x\" right too.
\n" ); document.write( "If you are lucky, you may \"see\" the right factors immediately.
\n" ); document.write( "If not, there is a procedure.
\n" ); document.write( "Multiply the coefficient of \"x%5E2\" times the independent (constant) term, in this case, \"6%2A%28-21%29=-126\"
\n" ); document.write( "Look for pairs of factors for that product
\n" ); document.write( "-126=-(1)(126)=-(2)(63)=-(3)(42)=-(6)(21)=-(7)(18)=-(9)(14)
\n" ); document.write( "Find a pair of factors that adds to the coefficient of x. In this case, -14 and 9 work, because -14+9=-5).
\n" ); document.write( "Re-write the polynomial with two terms in x with the coefficients found.
\n" ); document.write( "\"6x%5E2-5x-21=6x%5E2-14x%2B9x-21\"
\n" ); document.write( "Group the 4 terms in pairs with common factors.
\n" ); document.write( "\"6x%5E2-14x%2B9x-21=%286x%5E2%2B9x%29%2B%28-14x-21%29\"
\n" ); document.write( "Extract common factors, twice:
\n" ); document.write( "
\n" ); document.write( "So \"12x%5E2-10x-42=0\" is equivalent to \"6x%5E2-5x-21=0\" and is equivalent to \"%283x-7%29%282x%2B3%29=0\"
\n" ); document.write( "The solutions are what makes one or both factors zero:
\n" ); document.write( "\"3x-7=0\" --> \"3x=7\" --> \"highlight%28x=7%2F3%29\"
\n" ); document.write( "\"2x%2B3=0\" --> \"2x=-3\" --> \"highlight%28x=-3%2F2%29\"
\n" ); document.write( "QUADRATIC FORMULA (complicated, but always works)
\n" ); document.write( "An equation of the form \"ax%5E2%2Bbx%2Bc=0\" has the solutions \"x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\".
\n" ); document.write( "In this case a=6, b=-5, and c=-21, so
\n" ); document.write( " --> \"x=%285%2B23%29%2F12=28%2F12=highlight%287%2F3%29\" or
\n" ); document.write( "\"x=%285-23%29%2F12=-18%2F12=highlight%28-3%2F2%29\"
\n" ); document.write( "COMPLETING THE SQUARE
\n" ); document.write( "Transform the equation so that only the terms in \"x\"} and \"x%5E2\" are on one side of the equal sign, and the term in \"x%5E2\" is just \"x%5E2\".
\n" ); document.write( "\"6x%5E2-5x-21=0\" --> \"6x%5E2-5x=21\" --> \"x%5E2-5%2F6x=21%2F6\" --> \"x%5E2-5%2F6x=7%2F2\"
\n" ); document.write( "Add to both sides the number that will make the left side a perfect square, and then write it as a square, and simplify:
\n" ); document.write( "\"x%5E2-5%2F6x%2B25%2F144=7%2F2%2B25%2F144\" --> \"%28x-5%2F12%29%5E2=7%2A62%2F%282%2A62%29%2B25%2F144\" --> \"%28x-5%2F12%29%5E2=504%2F144%2B25%2F144\" --> \"%28x-5%2F12%29%5E2=529%2F144\" --> \"%28x-5%2F12%29%5E2=%2823%2F12%29%5E2\"
\n" ); document.write( "If \"%28x-5%2F12%29%5E2=%2823%2F12%29%5E2\", either \"x-5%2F12=23%2F12\" or \"x-5%2F12=-23%2F12\"
\n" ); document.write( "\"x-5%2F12=23%2F12\" -->\"x=%285%2B23%29%2F12=28%2F12=highlight%287%2F3%29\"
\n" ); document.write( "\"x-5%2F12=-23%2F12\" -->\"x=%285-23%29%2F12=-18%2F12=highlight%28-3%2F2%29\" \n" ); document.write( "
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