Solving Quadratic Equations by Completing the Square 07:00 minutes

Video Transcript

Transcript Solving Quadratic Equations by Completing the Square

Kata Ana is always late for ninja school. Luckily, she’s had a lot of practice finding the quickest way to her school. Kata needs to know how to solve quadratic equations by completing the square in order to avoid some of the dangerous obstacles on her way to school. Kata has to use her grappling hook to swing over to the lantern over there. She throws her grappling hook onto the school’s roof. Kata knows that she has to avoid the school's guard Komodo dragon. She uses a technique passed down for generations called completing the square.

Kata knows that if she does this, she can find the zeros of any parabola as well as the parabola’s vertex by putting the standard form of the quadratic formula, ax² + bx + c = 0 into vertex form, a(x - h)² + k = 0. Completing the Square certainly is a powerful tool.

Vertex Form

The first equation Kata has is one-eighth x² plus five over two 'x' plus nine over two equals 0. Kata looks at this problem. She tries to list the factors of negative nine over two: 1, negative nine over two and -1, nine over two. But neither of these options sum to five over two. Kata needs a more powerful tool. To complete the square, Kata needs to get the equation in vertex form. First, she moves the nine over two to the right hand side using opposite operations. This leaves her with one-eighth x² plus five over two 'x' equals negative nine over two.
Next, she removes one-eighth from the equation by dividing each coefficient by one-eighth, leaving her with one-eighth times the quantity 'x' squared plus twenty 'x' equals negative nine over two. Now she takes half of her ‘b’ term, 20, giving her 10. She then squares the 10 and needs to add the result on both sides of the equation. However, since this part of the equation is inside the parentheses, and the contents of the parentheses is multiplied by one-eighth, we have to add 100 times one-eighth on the right side of the equation. We should simplify the fractions on the right side of the equation as much as possible.

Kata notices that the left side of the equation is now a perfect square trinomial, which she knows how to factor. All Kata has to do now is move the 8 to the left side of the equation and compare her result to the standard vertex form of the equation to figure out the signs of the vertex. One-eighth times (x + 10)² minus 8= 0.Since the signs inside the parentheses don't match, we have to change 10 to negative 10. The signs are also different for the constant, so we have to change 8 to negative 8. The vertex has revealed itself! The vertex is (-10, -8)!

Finding the Zeros by Isolating x

Now, to get the zeros of the parabola! Let's start by getting rid of the one-eighth on the left side of the equation. We can do this by multiplying both sides of the equation by 8, giving us 'x' plus 10 quantity squared equals 64. Now we can take the square root of both sides. We're left with 'x' plus 10 equals plus/minus 8. For the last step, we isolate the 'x' and get our solutions, negative 2 and negative 18!
Completing the Square really is a powerful tool. Kata Ana can clearly see why her ancestors have been using it for generations!

Second Example

Kata now needs to be able to clear that wall without hitting the roof so she can land safely on the ledge of her school. The equation for this part is negative one-tenth x² + eight-fifths 'x' plus 18-fifths equals zero. For this equation, Kata Ana repeats the process she used to solve the first equation. She first moves the constant to the right-hand side. Then, she divides the left side of the equation by negative one-tenth.
Again, she takes half of her ‘b’ term, -16, giving her -8, which she then squares and adds to both sides of the equation. She remembers that, since the negative one-tenth is outside the parentheses, she has to multiply 64 by one-tenth before adding it to the right side of the equation. The left side of the equation is a perfect square trinomial! Completing the Square never fails!
Using opposite operations, Kata rewrites the left side of the equation as negative one-tenth times the quantity 'x' minus 8, squared plus 10. Great, now the equation is in standard form. Once again, we compare the signs. Can you tell what the vertex is? That's right! it's at (8, 10)! Once again, to get the zeros of the parabola! Taking the square root on both sides of the equation leaves 'x' minus eight equal to plus/minus 10. And finally, isolating the variable gives us our solutions. Thanks to her knowledge of completing the square and some quick thinking, Kata is able to land safely on the ledge of her school.

What’s this?! What's happened to her classmates?! Kata Ana is Shell Shocked!