{"id":138,"date":"2020-12-02T04:00:39","date_gmt":"2020-12-02T04:00:39","guid":{"rendered":"https:\/\/mathdynamics.net\/blog\/?p=138"},"modified":"2021-10-28T21:27:44","modified_gmt":"2021-10-28T21:27:44","slug":"a-quadratic-deep-dive-into-real-roots","status":"publish","type":"post","link":"https:\/\/mathdynamics.net\/blog\/index.php\/2020\/12\/02\/a-quadratic-deep-dive-into-real-roots\/","title":{"rendered":"A Quadratic Deep Dive into Real Roots"},"content":{"rendered":"\n

The Math Dynamics Multivariable Algebra Expression Based Graphing Calculator<\/span><\/p>\n\n\n\n

\n
\n
\"Graph
This Math Dynamics Function Card will generate a plot from -10 to 10 in increments of .001 and limit the Y axis from 10 to -10<\/figcaption><\/div><\/figure><\/div>\n<\/div>\n\n\n\n
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Complete the Square method is an algorithm to find the Roots of a Quadratic Equation both Real and Complex.\u00a0 This exercise focuses on using the “Complete the Square” method to calculate the Real Roots so it only makes sense to use a Quadratic Function that has two real roots.<\/p>\n<\/div>\n<\/div>\n\n\n\n

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We start with a Quadratic Function defined as:<\/p>\n\n\n\n

\\( -3*x^2 + 2*x + 4 \\)<\/strong><\/p>\n\n\n\n

The plot of this function visually demonstrates that this Quadratic Function has 2 real roots.  <\/p>\n\n\n\n

The exact values of these two real roots can be calculated using the “Complete the Square” method.  This requires two things:<\/p>\n\n\n\n

  1. \\( -3*x^2 + 2*x + 4 = 0\\)<\/strong><\/li>
  2. The \\(a*x^2\\)<\/strong> term requires that a = 1, not -3<\/li><\/ol>\n<\/div>\n\n\n\n
    \n
    \"\"<\/figure><\/div>\n<\/div>\n<\/div>\n\n\n\n
    \n
    \n
    \"Quadratic
    This Math Dynamics Function Card produces a Quadratic Equation that factors out the coefficient a ensuring it is always 1<\/figcaption><\/div><\/figure><\/div>\n<\/div>\n\n\n\n
    \n

    To resolve the second requirement requires that both sides of the quadratic equation be multiplied by -(1\/3) such that the new Quadratic Function is:<\/p>\n\n\n\n

    <\/p>\n\n\n\n

    \\( x^2 -(2\/3)*x -(4\/3) = 0\\)<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

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    \n

    Which produces a different graph.\u00a0 This new graph shares the same roots as the original graph.<\/p>\n<\/div>\n\n\n\n

    \n
    \"\"<\/figure><\/div>\n<\/div>\n<\/div>\n\n\n\n

    I found this to be a quite interesting property.\u00a0 It appears though that all multiples of a Quadratic Function have equal roots<\/p>\n\n\n\n

    \n
    \n
    \"Visual
    This Math Dynamics Card defines a function that can generate any multiple of any Quadratic Function<\/figcaption><\/div><\/figure><\/div>\n<\/div>\n\n\n\n
    \n
    \"\"<\/figure><\/div>\n<\/div>\n<\/div>\n\n\n\n
    \"Math
    Explore the Quadratic Equation and Using the Complete the Square Method to find the roots.<\/figcaption><\/div><\/a><\/figure>\n\n\n\n

    <\/p>\n","protected":false},"excerpt":{"rendered":"

    The Math Dynamics Multivariable Algebra Expression Based Graphing Calculator This Math Dynamics Function Card will generate a plot from -10 to 10 in increments of .001 and limit the Y axis from 10 to -10 Complete the Square method is an algorithm to find the Roots of a Quadratic Equation both Real and Complex.\u00a0 This […]<\/p>\n","protected":false},"author":1,"featured_media":134,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[17,16],"tags":[7,10,9,4,5,18],"_links":{"self":[{"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/posts\/138"}],"collection":[{"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/comments?post=138"}],"version-history":[{"count":70,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/posts\/138\/revisions"}],"predecessor-version":[{"id":397,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/posts\/138\/revisions\/397"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/media\/134"}],"wp:attachment":[{"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/media?parent=138"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/categories?post=138"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathdynamics.net\/blog\/index.php\/wp-json\/wp\/v2\/tags?post=138"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}