document.write( "Question 120039: Find the quadratic equation with roots -1+4i and -1-4i.
\n" ); document.write( "The correct answer is x^2 + 2x + 17 = 0 according to my teacher.
\n" ); document.write( "Is this answer wrong because everytime I try to solve the problem, I do not get that answer.
\n" ); document.write( "If it is right, please explain how. \n" ); document.write( "
Algebra.Com's Answer #87971 by bucky(2189)\"\" \"About 
You can put this solution on YOUR website!
You are given that the roots of a quadratic equation are:
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\n" ); document.write( "\"-1+%2B+4i\" and \"-1+-+4i\"
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\n" ); document.write( "and you are asked to find the quadratic equation that has those roots.
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\n" ); document.write( "Since these are roots, you know that they are the values of x that solve a quadratic
\n" ); document.write( "expression that is equal to zero. Therefore, you know that:
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\n" ); document.write( "\"x+=+-1+%2B+4i\"
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\n" ); document.write( "is one answer and
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\n" ); document.write( "\"x+=+-1+-+4i\"
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\n" ); document.write( "is the other answer.
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\n" ); document.write( "You can convert these two answers to the factors of the quadratic by getting everything
\n" ); document.write( "on the right side of these two equalities. So let's start with:
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\n" ); document.write( "\"x+=+-1+%2B+4i\"
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\n" ); document.write( "and subtract \"+-1+%2B+4i\" from both sides. When you do that subtraction, you get:
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\n" ); document.write( "\"x+%2B+1+-+4i+=+0\"
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\n" ); document.write( "This tells you that (x + 1 - 4i) is one of the factors of the quadratic.
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\n" ); document.write( "Similarly, next go to the other root:
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\n" ); document.write( "\"x+=+-1+-+4i\"
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\n" ); document.write( "Convert it to a factor by subtracting \"-1+-+4i\" from both sides. When you do that subtraction
\n" ); document.write( "you get:
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\n" ); document.write( "\"x+%2B+1+%2B+4i+=+0\"
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\n" ); document.write( "and this tells you that (x + 1 + 4i) is the other factor of the quadratic.
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\n" ); document.write( "To get the quadratic equation, multiply these two factors together and set the product equal
\n" ); document.write( "to zero.
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\n" ); document.write( "So you multiply (x + 1 - 4i) times (x + 1 + 4i) and set the result equal to zero.
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\n" ); document.write( "This multiplication requires that you multiply each term in the second set of parentheses
\n" ); document.write( "by each term in the first set of parentheses and then combine terms. So let's take the x
\n" ); document.write( "from the first set of parentheses and multiply it times everything in the second set of parentheses.
\n" ); document.write( "In other words multiply:
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\n" ); document.write( "\"x%2A%28x+%2B+1+%2B+4i%29\" to get \"x%5E2+%2B+x+%2B+4xi\"
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\n" ); document.write( "Next take the +1 from the first set of parentheses and multiply it times everything in
\n" ); document.write( "the second set of parentheses:
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\n" ); document.write( "\"1%2A%28x+%2B+1+%2B+4i%29\" to get \"x+%2B+1+%2B+4i\"
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\n" ); document.write( "Finally take the -4i from the first set of parentheses and multiply it times everything in
\n" ); document.write( "the second set of parentheses:
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\n" ); document.write( "\"-4i%2A%28x+%2B+1+%2B+4i%29\" to get \"-4xi+-+4i+-+16i%5E2\"
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\n" ); document.write( "In the third term of this last product, note that by definition \"i%5E2+=+-1\" and if in
\n" ); document.write( "\"-16i%5E2\" you replace \"i%5E2\" by -1 you get \"-16%28-1%29+=+%2B16\". So the third multiplication
\n" ); document.write( "for which you got \"-4xi+-+4i+-+16i%5E2+\" can be simplified to \"-4xi+-+4i+%2B+16\"
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\n" ); document.write( "Now combine all three of the products by adding them:
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\n" ); document.write( "\"%28x%5E2+%2B+x+%2B+4xi%29%2B%28x+%2B+1+%2B+4i%29%2B%28-4xi+-+4i+%2B+16%29\"
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\n" ); document.write( "Group the common terms and you get:
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\n" ); document.write( "\"x%5E2+%2B+%28x%2Bx%29+%2B+%284xi+-+4xi%29+%2B+%284i+-+4i%29+%2B+%281+%2B+16%29\"
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\n" ); document.write( "And when you combine the grouped terms some of them cancel out. You are left with:
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\n" ); document.write( "\"x%5E2+%2B+2x+%2B+17\"
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\n" ); document.write( "In the last step you set this result equal to zero and you have as the quadratic equation:
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\n" ); document.write( "\"x%5E2+%2B+2x+%2B+17+=+0\"
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\n" ); document.write( "Hope this helps you to find your error. In this case your teacher is correct.
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