## Subjects

# Solving quadratic equations by completing the square

What number do you add to both sides of the equation when completing the square of $x^2 + 10x = 7$?

## Solving quadratic equations by completing the square

There are different ways of solving quadratic equations. Here's one that is pretty elegant. If you take the time to really understand it, you'll be able to solve all quadratic equations without having to use any formula. Look at this equation. 'X-squared plus 'four x' minus 'five' equals 'zero'. Before we use this this method we'll need to tidy up the equation.

We want to gather all the terms containing x on the left side, and we want the constant term on the right. We now have 'x-squared' plus 'four x' on the left -- And five on the right. Now the equation is in a form we can use. This 'x-squared' plus 'four x' that's on the left: What does it really mean? One way to understand this is to illustrate the equation, piece by piece.

First, we have 'x-squared'. We can show that as a square, with each side being x. Its area is 'x times x': in other words 'x-squared'. Then there is the term 'four x'. This would be a rectangle with the sides x and four.

The area, the base times the height, is 'four times x'. If we add the square to the rectangle -- We get a new rectangle with the area 'x-squared' plus 'four x'. And 'x-squared' plus 'four x' equals five. Compare this to the equation. 'X-squared' plus 'four x' equals five. Now here's the clever thing.

We grab a pair of scissors and cut the small rectangle in half. We get two thin rectangles, that both have the area 'two x'. Then we move one of them and place it here. We haven't removed anything, we haven't added anything. We've just moved the shapes around.

So this figure has the same area as it had initially. The area is still five. The new figure looks almost like a square -- But only almost. There is a tiny part missing in the top right corner. What's missing?

A small square -- with the side -- two! We can use this small square to complete the big one. This method is called 'completing the square'. So what is the area of this new larger square? First we had five, and then we completed with two times two.

The area is five plus four, which equals nine. But hang on! There is another way of describing the area of a square: the side... squared! Each side is 'x plus two', so the area is 'x plus two' squared.

And we know already that this equals nine. By completing the square we have rewritten the equation we had initially. 'X-squared' plus 'four x' minus 'five' equals 'zero' -- To 'x' plus 'two-squared' equals 'nine'. There, we've done the hard part. Now we're going to solve the equation. And when we solve an equation using the square root, there are two possible solutions.

So 'x plus two' equals 'plus-or-minus the square root of nine', which is 'plus-or-minus three'. 'X -one' plus 'two' equals 'plus three'. 'X-two' plus 'two' equals 'minus three'. We have two solutions to the equation. 'X' equals 'one', And 'x' equals 'minus five'. So we have solved a quadratic equation by 'completing the square'. You don't have to draw and move rectangles over your paper. We can use this method anyway.

Check it out. Now we're going to solve the equation 'Two x-squared' Minus 'eight x' Plus 'six' Equals 'zero'. Let's do it step by step. Pause the film if you need extra time to think. Step one.

The coefficient in front of 'x-squared' has to be 'one', so we we divide the whole expression by 'two'. We get 'x-squared' minus 'four-x' plus 'three' equals 'zero'. Step two. Gather all terms with an x on the left side, and the constant term on the right side. We rewrite the equation to 'x-squared' minus 'four-x' equals 'minus three'.

Step three. Take the coefficient in front of the x-term, in this case 'minus four' we're going to cut it in half, in other words, divide by two. The result is minus two. Step four, the actual completion. First we square 'minus two' and get 'four'.

Add this to both sides of the equals sign. We get 'x-squared' minus 'four x' plus 'four' equals 'minus three' plus 'four' equals 'one'. And look! This expression looks familiar. We can use the rule of 'squaring a binomial'.

And go to -- step five. Now we write the equation as 'x' minus 'two-squared', which equals 'one'. 'X minus two' then is, 'plus-or-minus the square root of one', which is 'plus-or-minus one'. 'x-one' equals 'plus one, plus two', equals 'three'. 'x-two' equals 'minus one, plus two', equals 'one'. And we have solved the equation by, yes, completing the square.