document.write( "Question 462262: 1. Rewrite y = x2 + 14x + 29 in general form.
\n" ); document.write( "2. Rewrite y = 3x2 - 24x + 10 in general form.
\n" ); document.write( "3. Solve for x: (x - 9)2 = 1
\n" ); document.write( "4. Solve for x: x2 + 24x + 90 = 0
\n" ); document.write( "5. Solve for x: 2x2 - 4x - 14 = 0
\n" ); document.write( "6. Create your own quadratic equation and demonstrate how it would be solved by graphing, factoring, the quadratic formula, and by completing the square.
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Algebra.Com's Answer #316922 by MathLover1(20849)\"\" \"About 
You can put this solution on YOUR website!
first: Quadratic Functions in General Form is \"f%28x%29+=+ax2+%2B+bx+%2B+c\" or \"y+=+ax2+%2B+bx+%2B+c\"\r
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\n" ); document.write( "\n" ); document.write( "1.\"y+=+x%5E2+%2B+14x+%2B+29\"...already in general form.
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\n" ); document.write( "2. \"y+=+3x%5E2+-+24x+%2B+10\"...already in general form. \r
\n" ); document.write( "\n" ); document.write( "3. Solve for \"x\": \r
\n" ); document.write( "\n" ); document.write( "\"%28x+-+9%29%5E2+=+1\" \r
\n" ); document.write( "\n" ); document.write( "\"%28sqrt%28x+-+9%29%5E2%29+=+sqrt%281%29\"\r
\n" ); document.write( "\n" ); document.write( "\"x+-+9+=+1\"\r
\n" ); document.write( "\n" ); document.write( "\"x+=+1%2B9\"\r
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\n" ); document.write( "\n" ); document.write( "\"x+=+10\"\r
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\n" ); document.write( "\n" ); document.write( "4. Solve for \"x\": \r
\n" ); document.write( "\n" ); document.write( "\"x%5E2+%2B+24x+%2B+90+=+0+\"....use quadratic formula\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 \"x%5E2%2B24%2Ax%2B90=0\" ( notice \"a=1\", \"b=24\", and \"c=90\")
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\n" ); document.write( " \"x+=+%28-24+%2B-+sqrt%28+%2824%29%5E2-4%2A1%2A90+%29%29%2F%282%2A1%29\" Plug in a=1, b=24, and c=90
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\n" ); document.write( " \"x+=+%28-24+%2B-+sqrt%28+576-4%2A1%2A90+%29%29%2F%282%2A1%29\" Square 24 to get 576
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\n" ); document.write( " \"x+=+%28-24+%2B-+sqrt%28+576%2B-360+%29%29%2F%282%2A1%29\" Multiply \"-4%2A90%2A1\" to get \"-360\"
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\n" ); document.write( " \"x+=+%28-24+%2B-+sqrt%28+216+%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+=+%28-24+%2B-+6%2Asqrt%286%29%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+=+%28-24+%2B-+6%2Asqrt%286%29%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( "
\n" ); document.write( " \"x+=+%28-24+%2B+6%2Asqrt%286%29%29%2F2\" or \"x+=+%28-24+-+6%2Asqrt%286%29%29%2F2\"
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\n" ); document.write( " Now break up the fraction
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\n" ); document.write( " \"x=-24%2F2%2B6%2Asqrt%286%29%2F2\" or \"x=-24%2F2-6%2Asqrt%286%29%2F2\"
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\n" ); document.write( " Simplify
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\n" ); document.write( " \"x=-12%2B3%2Asqrt%286%29\" or \"x=-12-3%2Asqrt%286%29\"
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\n" ); document.write( " So the solutions are:
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\n" ); document.write( " \"x=-12%2B3%2Asqrt%286%29\" or \"x=-12-3%2Asqrt%286%29\"
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\n" ); document.write( "\n" ); document.write( "5. Solve for \"x\": \r
\n" ); document.write( "\n" ); document.write( "\"2x%5E2+-+4x+-+14+=+0\"....use quadratic formula\r
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Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:

\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( " Starting with the general quadratic
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\n" ); document.write( "
\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 \"2%2Ax%5E2-4%2Ax-14=0\" ( notice \"a=2\", \"b=-4\", and \"c=-14\")
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\n" ); document.write( " \"x+=+%28--4+%2B-+sqrt%28+%28-4%29%5E2-4%2A2%2A-14+%29%29%2F%282%2A2%29\" Plug in a=2, b=-4, and c=-14
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\n" ); document.write( " \"x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%2A2%2A-14+%29%29%2F%282%2A2%29\" Negate -4 to get 4
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\n" ); document.write( " \"x+=+%284+%2B-+sqrt%28+16-4%2A2%2A-14+%29%29%2F%282%2A2%29\" Square -4 to get 16 (note: remember when you square -4, you must square the negative as well. This is because \"%28-4%29%5E2=-4%2A-4=16\".)
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\n" ); document.write( " \"x+=+%284+%2B-+sqrt%28+16%2B112+%29%29%2F%282%2A2%29\" Multiply \"-4%2A-14%2A2\" to get \"112\"
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\n" ); document.write( " \"x+=+%284+%2B-+sqrt%28+128+%29%29%2F%282%2A2%29\" Combine like terms in the radicand (everything under the square root)
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\n" ); document.write( " \"x+=+%284+%2B-+8%2Asqrt%282%29%29%2F%282%2A2%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+=+%284+%2B-+8%2Asqrt%282%29%29%2F4\" Multiply 2 and 2 to get 4
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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+=+%284+%2B+8%2Asqrt%282%29%29%2F4\" or \"x+=+%284+-+8%2Asqrt%282%29%29%2F4\"
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\n" ); document.write( " Now break up the fraction
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\n" ); document.write( " \"x=%2B4%2F4%2B8%2Asqrt%282%29%2F4\" or \"x=%2B4%2F4-8%2Asqrt%282%29%2F4\"
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\n" ); document.write( " Simplify
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\n" ); document.write( " \"x=1%2B2%2Asqrt%282%29\" or \"x=1-2%2Asqrt%282%29\"
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\n" ); document.write( " So the solutions are:
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\n" ); document.write( " \"x=1%2B2%2Asqrt%282%29\" or \"x=1-2%2Asqrt%282%29\"
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\n" ); document.write( "\n" ); document.write( "6. Create your own quadratic equation and demonstrate how it would be solved by graphing, factoring, the quadratic formula, and by completing the square. \r
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\n" ); document.write( "\n" ); document.write( "\"y=x%5E2+%2B+3x+%2B+2\"....use quadratic formula\r
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Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:

\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( " Starting with the general quadratic
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\n" ); document.write( "
\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%2B2=0\" ( notice \"a=1\", \"b=3\", and \"c=2\")
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\n" ); document.write( " \"x+=+%28-3+%2B-+sqrt%28+%283%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29\" Plug in a=1, b=3, and c=2
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\n" ); document.write( " \"x+=+%28-3+%2B-+sqrt%28+9-4%2A1%2A2+%29%29%2F%282%2A1%29\" Square 3 to get 9
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\n" ); document.write( "
\n" ); document.write( " \"x+=+%28-3+%2B-+sqrt%28+9%2B-8+%29%29%2F%282%2A1%29\" Multiply \"-4%2A2%2A1\" to get \"-8\"
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\n" ); document.write( "
\n" ); document.write( " \"x+=+%28-3+%2B-+sqrt%28+1+%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-+1%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-+1%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( "
\n" ); document.write( " \"x+=+%28-3+%2B+1%29%2F2\" or \"x+=+%28-3+-+1%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+1%29%2F2\"
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\n" ); document.write( " \"x=-2%2F2\" 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=%28-3+-+1%29%2F2\"
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\n" ); document.write( " \"x=-4%2F2\" Subtract the terms in the numerator
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\n" ); document.write( " \"x=-2\" Divide
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\n" ); document.write( " So another answer is
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\n" ); document.write( " \"x=-2\"
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\n" ); document.write( " So our solutions are:
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\n" ); document.write( " \"x=-1\" or \"x=-2\"
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\n" ); document.write( "\n" ); document.write( "\"y=x%5E2+%2B+3x+%2B+2\"....use graphing\r
\n" ); document.write( "\n" ); document.write( "find first several points that lie on a parabola\r
\n" ); document.write( "\n" ); document.write( "if \"x=0\"...->...\"y=2\"\r
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\n" ); document.write( "\n" ); document.write( "if \"x=1\"...->...\"y=6\"\r
\n" ); document.write( "\n" ); document.write( "if \"x=-1\"...->...\"y=0\"\r
\n" ); document.write( "\n" ); document.write( "if \"x=-2\"...->...\"y=0\"\r
\n" ); document.write( "\n" ); document.write( "if \"x=-3\"...->...\"y=2\"\r
\n" ); document.write( "\n" ); document.write( "\"+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2+%2B+3x+%2B+2%29+\"\r
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\n" ); document.write( "\n" ); document.write( "\"y=x%5E2+%2B+3x+%2B+2\"....use factoring\r
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Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor \"1%2Ax%5E2%2B3%2Ax%2B2\", first we need to ask ourselves: What two numbers multiply to 2 and add to 3? Lets find out by listing all of the possible factors of 2
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\n" ); document.write( " Factors:
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\n" ); document.write( " 1,2,
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\n" ); document.write( " -1,-2,List the negative factors as well. This will allow us to find all possible combinations
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\n" ); document.write( " These factors pair up to multiply to 2.
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\n" ); document.write( " 1*2=2
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\n" ); document.write( " (-1)*(-2)=2
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\n" ); document.write( " note: remember two negative numbers multiplied together make a positive number
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\n" ); document.write( " Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3
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First Number|Second Number|Sum
1|2|1+2=3
-1|-2|-1+(-2)=-3
We can see from the table that 1 and 2 add to 3. So the two numbers that multiply to 2 and add to 3 are: 1 and 2\r\n" ); document.write( " \r\n" ); document.write( " Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:\r\n" ); document.write( " \r\n" ); document.write( " \"%28x%2Ba%29%28x%2Bb%29\"substitute a=1 and b=2\r\n" ); document.write( " \r\n" ); document.write( " So the equation becomes:\r\n" ); document.write( " \r\n" ); document.write( " (x+1)(x+2)\r\n" ); document.write( " \r\n" ); document.write( " \r\n" ); document.write( " Notice that if we foil (x+1)(x+2) we get the quadratic \"1%2Ax%5E2%2B3%2Ax%2B2\" again\n" ); document.write( "

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\n" ); document.write( "\n" ); document.write( "\"y=x%5E2+%2B+3x+%2B+2\"....use completing the square\r
\n" ); document.write( "\n" ); document.write( "convert it into \"y=%281%2F5%29x%5E2%2B%281%2F2%29x%2B1%2F10\"\r
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