ShmoopTube

00:04

Solving Quadratic Equations: Completing the Square, a la Shmoop.

00:09

A new park in town is being funded by a conglomerate of handkerchief makers. [Man sneezes and a woman hands him a handkerchief]

00:14

They've named the park Pocket Square.

00:16

But they are having trouble completing the square, so they call... ['Pocket Square' sign drops into the park]

00:21

…The QuadSquad In order to complete the square, the QuadSquad ['The QuadSquad' car arrives]

00:25

has to solve this problem:

00:27

Two-x-squared plus five-x minus 42 equals zero. [Man stood in a hard hat next to a blu print sheet]

00:33

Okay to solve first, add forty-two to both sides to get...

00:39

…two-x-squared plus eight-x equals 42.

00:42

Divide every term by two so that the number before the x-squared is 1.

00:47

…x-squared plus four-x equals twenty-one.

00:51

It's time to complete the square. [Woman holding the equation on a sheet waiting expectantly]

00:53

We want to add a number to the left side such that it becomes a perfect square.

00:59

We can find this number by taking the coefficient of the x-term, dividing it by 2, and squaring

01:06

it.

01:07

In this case, our coefficient of the x term is 4.

01:11

4 divided by 2 equals 2.

01:15

Two-squared equals 4.

01:17

Eureka!

01:18

We've found the number that completes the square. [The answer is highlighted]

01:22

Whatever we do to one side, we have to do to the other.

01:26

So we add 4 to both sides, which gets us x squared plus 4x plus 4 equals 21 plus 4.

01:34

Now we can factor the left side of the equation. [The left side is highlighted]

01:38

X-squared plus four-x plus four can be simplified to x plus 2 in parentheses squared.

01:44

To simplify the right-hand side, we add 21 and 4 to get 25. [The right hand side is highlighted]

01:50

All that's left is to take the square root of both sides. [Both sides have a square root sign drawn over them]

01:54

Don't forget that the square root can be positive or negative. [Finger appears with piece of knotted string attached]

01:57

We're left with x plus 2 equals plus-or-minus the square-root of 25, or just 5.

02:04

Subtract two from both sides...

02:06

and we get two answers.

02:08

x equals either negative 2 plus 5..

02:11

…or negative 2 minus 5.. which gives us 3 and negative 7.

02:18

And there you have it! [Man in the hard hat points to the answer]

02:19

The equation for the square is complete.

02:21

And Pocket Square is complete as well. [The QuadSquad car reverses away]

Full Transcript