SOLUTION: The square of the differene between a number and 12 is twice the number. what is the number?

Question 496643: The square of the differene between a number and 12 is twice the number.
what is the number?
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Let x be the unknown number. Then the difference between the unknown number and 12 can be written as . This quantity is to be squared and it is to equal two times the unknown number. In equation form this becomes:
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Squaring out the left side makes the equation become:
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Subtract 2x from both sides to get the equation into the more conventional quadratic form having zero on the right side:
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This quadratic equation can be solved by factoring the left side as follows:
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And this equation will be true if either of the factors is zero because a multiplication on the left side by a zero factor makes the left side equal to the right side. So either:
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which by adding 18 to both sides becomes
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or:
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which by adding 8 to both sides becomes
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If you didn't recognize the factoring, you could always use the quadratic formula to solve the quadratic equation. Recognize that the quadratic equation that we developed is in the form:
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and by comparing this form to the equation we developed, you can see that a = 1, b = -26, and c = 144. Substitute these values into the quadratic formula as follows:
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Within the radical sign, square the -26 and multiply out the -4*1*144 to get:
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and the terms in the radical become 100. Taking the square root of the 100 reduces the formula to:
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Remove the parentheses around the -26 and also multiply out the denominator to get:
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and this becomes either or and from these we again get the answers:
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and
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Check: Does equal ? If you work this out you get so it does work.
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Also check: Does equal ? If you work this out you find that are equal and so this solution also works.
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The two answers to this problem that will satisfy the requirements are x = 18 and x = 8.
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Hope this helps you understand how to go about working this problem.
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