# Solving Quadratic Equations: Completing the Squares

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Why “quadratic?” Is 4 its lucky number? Or does it just spend a lot of time at the gym?

Language | English Language |

Mathematics and Statistics Assessment | Quadratic and Other Polynomial Expressions, and Functions |

Quadratic Equations | Quadratic Equations |

### Transcript

has to solve this problem:

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

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

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

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

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

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

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

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

it.

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

4 divided by 2 equals 2.

Two-squared equals 4.

Eureka!

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

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

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

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

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

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

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

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

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

Subtract two from both sides...

and we get two answers.

x equals either negative 2 plus 5..

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

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

The equation for the square is complete.

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