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Add exercise: Solutions to quadratic equations
Reviewers: mark Reviewed By: mark Differential Revision: http://phabricator.khanacademy.org/D569
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| <!DOCTYPE html> | ||
| <html data-require="math math-format graphie khanscript"> | ||
| <head> | ||
| <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> | ||
| <title>Solutions to quadratic equations</title> | ||
| <script src="../khan-exercise.js"></script> | ||
| </head> | ||
| <body> | ||
| <div class="exercise"> | ||
| <div class="problems"> | ||
| <div id="one-rational"> | ||
| <div class="vars"> | ||
| <var id="ANSWER">"One rational solution"</var> | ||
| <var id="B">randRangeNonZero(-6, 6) * 2</var> | ||
| <var id="A">randFromArray(getFactors(B * B / 4))</var> | ||
| <var id="C">(B * B) / (4 * A)</var> | ||
| <var id="DISCRIMINANT">B * B - 4 * A * C</var> | ||
| </div> | ||
|
|
||
| <p class="question"> | ||
| Describe the solutions to the following quadratic equation: | ||
| </p> | ||
| <div class="problem"> | ||
| <code><var> | ||
| expr(["+", | ||
| ["*", A, ["^", "x", 2]], | ||
| ["*", B, "x"], | ||
| C]) | ||
| </var> = 0</code> | ||
| </div> | ||
|
|
||
| <div class="solution"><var>ANSWER</var></div> | ||
| <ul class="choices" data-category="true"> | ||
| <li>One rational solution</li> | ||
| <li>Two rational solutions</li> | ||
| <li>Two irrational solutions</li> | ||
| <li>One complex solution</li> | ||
| <li>Two complex solutions</li> | ||
| </ul> | ||
|
|
||
| <div class="hints"> | ||
| <div> | ||
| <p> | ||
| We could use the quadratic formula to solve for the | ||
| solutions and see what they are, but there's a | ||
| shortcut... | ||
| </p> | ||
| <p><code id="quadratic">\qquad | ||
| x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} | ||
| </code> | ||
| </div> | ||
| <div> | ||
| <script type="text/khanscript"> | ||
| $("#quadratic").text("\\qquad x = \\dfrac{-b \\pm" + | ||
| " \\sqrt{\\blue{b^2 - 4ac}}}{2a}"); | ||
| MathJax.Hub.Queue(["Reprocess", MathJax.Hub, | ||
| $("#quadratic")[0]]); | ||
| </script> | ||
| <div> | ||
| Think about what the part of the quadratic | ||
| formula <span class="hint_blue">under the | ||
| radical</span> tells us about the solutions. | ||
| </div> | ||
| </div> | ||
| <div> | ||
| <p> | ||
| Substitute the <code>a</code>, <code>b</code>, and | ||
| <code>c</code> coefficients from the quadratic | ||
| equation: | ||
| </p> | ||
| <p> | ||
| <code>\qquad\begin{array} | ||
| && b^2-4ac \\ \\ | ||
| =& <var>B</var>^2 - 4 ( | ||
| <var>A</var>)(<var>C</var>) \\ \\ | ||
| =& <var>DISCRIMINANT</var> | ||
| \end{array} | ||
| </code> | ||
| </p> | ||
| </div> | ||
| <p data-if="ANSWER === 'One rational solution'"> | ||
| Because <code>\blue{b^2 - 4ac} = 0</code>, then the | ||
| quadratic formula reduces to | ||
| <code>\dfrac{-b}{2a}</code>, which means there | ||
| is just one rational solution. | ||
| </p> | ||
| <p data-else-if="ANSWER === 'Two complex solutions'"> | ||
| Because <code>\blue{b^2 - 4ac}</code> is negative, its | ||
| square is imaginary and the quadratic formula reduces to | ||
| <code>\dfrac{-b \pm \sqrt{<var>DISCRIMINANT</var>}}{2a} | ||
| </code>, which means there are two complex solutions. | ||
| </p> | ||
| <p data-else-if="ANSWER === 'Two rational solutions'"> | ||
| Because <code>\blue{b^2 - 4ac}</code> is a perfect | ||
| square, its square root is rational and the | ||
| quadratic formula reduces to | ||
| <code>\dfrac{-b \pm <var>sqrt(DISCRIMINANT)</var>}{2a} | ||
| </code>, which means there are two rational solutions. | ||
| </p> | ||
| <p data-else> | ||
| Because <code>\blue{b^2 - 4ac}</code> is not a perfect | ||
| square, its square root is irrational and the | ||
| quadratic formula reduces to | ||
| <code>\dfrac{-b \pm \sqrt{<var>DISCRIMINANT</var>}}{2a} | ||
| </code>, which means there are two irrational solutions. | ||
| </p> | ||
| </div> | ||
| </div> | ||
|
|
||
| <div id="two-rational" data-type="one-rational"> | ||
| <div class="vars" data-apply="replace"> | ||
| <var id="ANSWER">"Two rational solutions"</var> | ||
| <div data-ensure="DISCRIMINANT > 0 && | ||
| sqrt(DISCRIMINANT) - floor(sqrt(DISCRIMINANT)) < 0.001 | ||
| && DISCRIMINANT <= 144"> | ||
| <var id="A">randRangeNonZero(-9, 9)</var> | ||
| <var id="B">randRangeNonZero(-9, 9)</var> | ||
| <var id="C">randRangeNonZero(-9, 9)</var> | ||
| <var id="DISCRIMINANT">B * B - 4 * A * C</var> | ||
| </div> | ||
| </div> | ||
| </div> | ||
|
|
||
| <div id="two-irrational" data-type="one-rational"> | ||
| <div class="vars" data-apply="replace"> | ||
| <var id="ANSWER">"Two irrational solutions"</var> | ||
| <div data-ensure="DISCRIMINANT > 0 && | ||
| sqrt(DISCRIMINANT) - floor(sqrt(DISCRIMINANT)) > 0.001"> | ||
| <var id="A">randRangeNonZero(-9, 9)</var> | ||
| <var id="B">randRangeNonZero(-9, 9)</var> | ||
| <var id="C">randRangeNonZero(-9, 9)</var> | ||
| <var id="DISCRIMINANT">B * B - 4 * A * C</var> | ||
| </div> | ||
| </div> | ||
| </div> | ||
|
|
||
| <div id="two-complex" data-type="one-rational"> | ||
| <div class="vars" data-apply="replace"> | ||
| <var id="ANSWER">"Two complex solutions"</var> | ||
| <div data-ensure="DISCRIMINANT < 0"> | ||
| <var id="A">randRangeNonZero(-9, 9)</var> | ||
| <var id="B">randRangeNonZero(-9, 9)</var> | ||
| <var id="C">randRangeNonZero(-9, 9)</var> | ||
| <var id="DISCRIMINANT">B * B - 4 * A * C</var> | ||
| </div> | ||
| </div> | ||
| </div> | ||
| </div> | ||
| </div> | ||
| </body> | ||
| </html> |