{"id":159,"date":"2014-05-09T12:22:32","date_gmt":"2014-05-09T16:22:32","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=159"},"modified":"2014-05-09T12:22:32","modified_gmt":"2014-05-09T16:22:32","slug":"it-takes-a-square","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=159","title":{"rendered":"It Takes a Square…"},"content":{"rendered":"

…to Complete a Square<\/strong><\/p>\n

As you have seen, quadratic equations can pop up unexpectedly.\u00a0 Here\u2019s one:<\/p>\n

x2<\/sup> + 4x – 12 = 0<\/i><\/strong><\/p>\n

I chose this one because it does \u201cfactor\u201d, but I don\u2019t want to solve it that way.\u00a0 Factoring causes stress.\u00a0 When you have not yet found the factors, you start to wonder:<\/p>\n

Is it just me? Or does this thing have no factors?\u00a0 <\/i><\/p>\n

When you are ready to give up, you turn to the quadratic formula.\u00a0 To solve:<\/p>\n

ax2<\/sup> + bx + c = 0<\/i><\/strong><\/p>\n

you just have to identify the values of a, b and c and then plug them into the magic formula:<\/p>\n

\"post11pic4\"<\/a><\/p>\n

For our example, a=1, b=4 and c= -12.\u00a0 So we get:<\/p>\n

\"post11pic5\"<\/a><\/p>\n

But you have to admit, there\u2019s something mindless and robot-like about solving quadratics this way.\u00a0 It\u2019s like we are impersonating a computer.\u00a0 If you are going to be mindless, you might as well use the solve function on a TI-89 and be done with it.\u00a0 But there is another alternative.\u00a0 We could go back to the algorithm that the quadratic formula is derived from.\u00a0 It\u2019s clever and it has style.\u00a0 And even better, it has diagrams!<\/p>\n

COMPLETING THE SQUARE<\/i><\/strong><\/p>\n

Let\u2019s start again with:<\/p>\n

x2<\/sup> + 4x – 12 = 0<\/i><\/strong><\/p>\n

And re-write it:<\/p>\n

x2<\/sup> + 4x = 12<\/i><\/strong><\/p>\n

And one more time:<\/p>\n

x2<\/sup> + 2x +2x = 12<\/i><\/strong><\/p>\n

But why do this?\u00a0 Well, the left side of that equation has a nice picture that goes with it:<\/p>\n

\"post11pic1\"<\/a><\/p>\n

It\u2019s ALMOST a square.\u00a0 And if we add one little square\u2026<\/p>\n

\"post11pic2\"<\/a><\/p>\n

We have just \u201ccompleted the square\u201d!<\/p>\n

By adding that little blue square, we have made a bigger square with a side of length (x+2).<\/p>\n

But that addition would leave our original equation out of balance.\u00a0 So if we add 22<\/sup> to one side, we also have to add it to the other side:<\/p>\n

x2<\/sup> + 2x +2x + 4<\/span> = 12 + 4<\/span><\/i><\/strong><\/p>\n

As you can see from the picture, the left side of our equation is now a perfect square. That means that we can write:<\/p>\n

(x + 2)2<\/sup> = 16<\/i><\/strong><\/p>\n

We can solve this pretty quickly.\u00a0 Take the square root of both sides to get:<\/p>\n

(x + 2) = <\/i>\u00b14<\/i><\/strong><\/p>\n

x= -4 \u2013 2 = -6 or 4 \u2013 2 = 2<\/i><\/strong><\/i><\/b><\/p>\n

Q: Isn\u2019t factoring faster?\u00a0<\/i><\/b><\/p>\n

A: It is when it is.\u00a0 But when it isn\u2019t, it really isn\u2019t. And completing the square works either way.<\/i><\/b><\/p>\n

\u00a0<\/i>For example, here\u2019s one that doesn’t factor:<\/p>\n

x2<\/sup> + 5x – 4 = 0<\/i><\/strong><\/p>\n

x2<\/sup> + 2.5x +2.5x = 4<\/i><\/strong><\/p>\n

\"post11pic3\"<\/a>\u00a0<\/i><\/b><\/p>\n

<\/i><\/b>x2<\/sup> + 2.5x +2.5x + 2.52<\/sup><\/span>= 4 + 2.52<\/sup><\/span><\/i><\/strong><\/p>\n

(x + 2.5)2<\/sup> = 10.25<\/i><\/strong><\/p>\n

\u00a0\"post11pic6\"<\/a><\/i><\/b><\/p>\n

<\/i><\/b>You will still need a calculator if you want to tidy up, but you would have needed one had you used the quadratic formula instead.<\/p>\n

OK, I am not expecting to win a lot of converts here.\u00a0 I just think that when you connect it with its diagrams, completing the square has an elegance to it that the quadratic formula lacks. \u00a0(But if I had to do a whole page of these to do,say 1 \u2013 99 odd, I would grab a TI-89 and move on.)<\/p>\n

I have only seen one physics student use completing the square in class.\u00a0 She did it effortlessly with the same automaticity that most students bring to \u201ccross-multiply and divide\u201d.\u00a0 But she learned her algebra in another country.<\/p>\n","protected":false},"excerpt":{"rendered":"

…to Complete a Square As you have seen, quadratic equations can pop up unexpectedly.\u00a0 Here\u2019s one: x2 + 4x – 12 = 0 I chose this one because it does \u201cfactor\u201d, but I don\u2019t want to solve it that way.\u00a0 … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-2z","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/159"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=159"}],"version-history":[{"count":6,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/159\/revisions"}],"predecessor-version":[{"id":171,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/159\/revisions\/171"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=159"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=159"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=159"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}