Lesson video

Hi, I'm Miss.

Davies.

In this lesson, we're going to be writing quadratic expressions in the form of completing the square.

Squares or square numbers can be shown visually.

For example, one squared is one.

So we have one square by one square, so we have one square.

Two squared is two lengths across two lengths down.

This is four squares in total.

Three squared is nine.

And so on.

Squared algebraic expressions can be shown visually too.

This blue square has a length of x.

This means the area of this square is x squared.

What would x add one squared look like? Along the top we have an x length and one, and vertically we have a length of x and one.

In this square all together, we have x squared add 2x add one.

What about a subtract three squared? This would look like this.

Our length of the blue square is a and the three red rectangles have a length of negative three.

Altogether, we have got a squared subtract 6a add nine.

Consider the expression x squared add 6x add nine, and what it would look like with algebra tiles.

We can put this in the form of a square x squared add 6x add nine can be written as x add three squared.

This is known as completed square form.

The general term is x add a squared add b.

Let's look at x squared add 6x add eight.

These algebra tiles don't form a perfect square.

To go from x squared add 6x add nine to x squared add 6x add eight, we subtract one.

The completed square representation of x squared add 6x add eight is x add three squared subtract one.

Looking at x squared add 6x add 10 with algebra tiles, we can create a square with one leftover.

This square has a length of x add three.

To go from x add three squared to our x squared add 6x add 10, we add one.

This expression written in completed square form is x add three squared add one.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

To go from the expression x squared add 4x add four to the expression in part A, we add three.

To go from x squared add 4x add four to the expression in part B, we subtract one.

Look at the following examples and have a think about what is the same and what is different.

With these examples they have both got nine one's.

Our first example has six x's, and the second example has eight x's.

On this second example, both of our expressions have eight ones.

Our first example has 6x, and our second example has 4x.

With this last pair of equations, we can see that they both have 10 ones.

The first example has 6x, and the second example has 2x.

With all of these examples the groups of x's have been split into two equal groups.

If a quadratic is written in the form x squared add px add q, then it can be written in the form x add a squared add b.

This is where the value of a is the p-value or the x coefficient divided by two.

And the b-value is q, which is the constant subtract a squared.

If we look at the example, x squared add 6x add eight, this will be written as x add six divided by two, add eight, subtract six divided by two squared.

Six divided by two is three, and three squared is nine.

These can simplify to x add three squared subtract one.

Looking at the next example of x squared add 2x add 11.

If we split our x coefficient into two equal parts that gives us one.

We can then calculate 11 subtract one squared.

This gives us an answer of x add one squared add 10.

You can see from the algebra tiles that there are 10 ones leftover from this square.

Let's have a look at these next examples.

With the first expression to find the value of a we're going to divide the x coefficient by two.

10 divided by two is five.

We can then square five to get 25.

20 subtract 25 is negative five.

With our second example, we're going to divide the x coefficient by two.

This gives us 5/2.

We can then square this and subtract it from 11.

This gives us x add 5/2 squared, add 11, subtract 25/4.

This simplifies to give a solution of x add 5/2 squared, add 19/4.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

It is easier to leave any non-integers in fractional form rather than converting them into decimals.

In this next example, we're going to write the following quadratics in completed square form, which can be written as x add a squared add b.

First, we're going to represent x squared, subtract 2x add 11, with algebra tiles.

To start forming these as a square we've got x squared subtract 2x add one.

This square has a length of x subtract one.

x subtract one can be written as x subtract two divided by two squared, as it is a square, add 11 subtract negative 2/2 squared.

Two divided by two is one.

So we can rewrite this as x subtract one squared, add 11, subtract negative one squared.

Negative one squared is one.

11 subtract one simplifies to 10.

This expression is now written in the correct form.

Our next example is x squared subtract 3/2 x add one.

We're going to start off by dividing the x coefficient by two.

This is written as x subtract 3/2 divided by two squared, add one, subtract negative 3/2 divided by two squared.

3/2 divided by two is 3/4.

3/4 squared is 9/16.

One subtract 9/16 can be rewritten as 16/16 subtract 9/16.

This simplifies to give negative 7/16.

This is now written in the correct completed square form.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

Remember, that a negative number squared, gives a positive result.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

The value of a is 1/2 the x coefficient.

The value of b is the constant subtract a squared.

In this case, that's 7p subtract 2p squared.

That's all for this lesson.

Thanks for watching.