# mwittels/khan-exercises forked from Khan/khan-exercises

Reviewers: mark

Reviewed By: mark

Differential Revision: http://phabricator.khanacademy.org/D569
beneater committed Aug 10, 2012
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 @@ -0,0 +1,151 @@ + + + + + Solutions to quadratic equations + + + +
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+ "One rational solution" + randRangeNonZero(-6, 6) * 2 + randFromArray(getFactors(B * B / 4)) + (B * B) / (4 * A) + B * B - 4 * A * C +
+ +

+ Describe the solutions to the following quadratic equation: +

+
+ + expr(["+", + ["*", A, ["^", "x", 2]], + ["*", B, "x"], + C]) + = 0 +
+ +
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+
• One rational solution
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• Two rational solutions
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• Two irrational solutions
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• One complex solution
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• Two complex solutions
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+ We could use the quadratic formula to solve for the + solutions and see what they are, but there's a + shortcut... +

+

\qquad + x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} + +

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+ Substitute the a, b, and + c coefficients from the quadratic + equation: +

+

+ \qquad\begin{array} + && b^2-4ac \\ \\ + =& B^2 - 4 ( + A)(C) \\ \\ + =& DISCRIMINANT + \end{array} + +

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+

+ Because \blue{b^2 - 4ac} = 0, then the + quadratic formula reduces to + \dfrac{-b}{2a}, which means there + is just one rational solution. +

+

+ Because \blue{b^2 - 4ac} is negative, its + square is imaginary and the quadratic formula reduces to + \dfrac{-b \pm \sqrt{DISCRIMINANT}}{2a} + , which means there are two complex solutions. +

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+ Because \blue{b^2 - 4ac} is a perfect + square, its square root is rational and the + quadratic formula reduces to + \dfrac{-b \pm sqrt(DISCRIMINANT)}{2a} + , which means there are two rational solutions. +

+

+ Because \blue{b^2 - 4ac} is not a perfect + square, its square root is irrational and the + quadratic formula reduces to + \dfrac{-b \pm \sqrt{DISCRIMINANT}}{2a} + , which means there are two irrational solutions. +

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+ "Two rational solutions" +
+ randRangeNonZero(-9, 9) + randRangeNonZero(-9, 9) + randRangeNonZero(-9, 9) + B * B - 4 * A * C +
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+ +
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+ "Two irrational solutions" +
+ randRangeNonZero(-9, 9) + randRangeNonZero(-9, 9) + randRangeNonZero(-9, 9) + B * B - 4 * A * C +
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+ +
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+ "Two complex solutions" +
+ randRangeNonZero(-9, 9) + randRangeNonZero(-9, 9) + randRangeNonZero(-9, 9) + B * B - 4 * A * C +
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