# Lesson 5

Squares and Circles

- Let’s see how the distributive property can relate to equations of circles.

### 5.1: Math Talk: Distribution

Distribute each expression mentally.

\(5(x+3)\)

\(x(x-3)\)

\((x+4)(x+2)\)

\((x-5)(x-5)\)

### 5.2: Perfectly Square

- Apply the distributive property to each expression.
- \((x-7)(x-7)\)
- \((x+4)^2\)
- \((x-10)^2\)
- \((x+1)^2\)

- Look at your results. Each of these expressions is called a
*perfect square trinomial*. Why? - Which of these expressions are perfect square trinomials? If you get stuck, look for patterns in your earlier work.
- \(x^2-6x+9\)
- \(x^2+10x+20\)
- \(x^2+18x+81\)
- \(x^2-2x+1\)
- \(x^2+4x+16\)

- Rewrite the perfect square trinomials you identified as squared binomials.

### 5.3: Back and Forth

- Here is the equation of a circle: \((x-2)^2+(y+7)^2=10^2\)
- What are the center and radius of the circle?
- Apply the distributive property to the squared binomials and rearrange the equation so that one side is 0. This is the form in which many circle equations are written.

- This equation looks different, but also represents a circle: \(x^2+6x+9+y^2-10y+25=64\)
- How can you rewrite this equation to find the center and radius of the circle?
- What are the center and radius of the circle?

In three-dimensional space, there are 3 coordinate axes, called the \(x\)-axis, the \(y\)-axis, and the \(z\)-axis. Write an equation for a sphere with center \((a,b,c)\) and radius \(r\).

### Summary

Suppose we square several binomials, or expressions that contain 2 terms. We get trinomials, or expressions that contain 3 terms. Does any pattern emerge in the results?

\((x+6)^2=x^2+12x+36\)

\((x-8)^2=x^2-16x+64\)

\((x+5)^2=x^2+10x+25\)

Each of the expressions on the right are called perfect square trinomials because they are the result of multiplying an expression by itself. There is a pattern in the results: When the coefficient of \(x^2\) in a trinomial is 1, if the constant term is the square of half the coefficient of \(x\), then the expression is a perfect square trinomial.

For example, \(x^2-14x+49\) is a perfect square trinomial because the constant term, 49, can be rewritten as (-7)^{2}, and half of -14 is -7. This expression can be rewritten as a squared binomial: \((x-7)^2\).

Two squared binomials show up in the equation for circles: \((x-h)^2+(y-k)^2=r^2\). Equations for circles are sometimes written in different forms, but we can rearrange them to help find the center and radius of the circle. For example, suppose the equation of a circle is written like this:

\(x^2-22x+121+y^2+2y+1=225\)

We can’t immediately identify the center and radius of the circle. However, if we rewrite the two perfect square trinomials as squared binomials and rewrite the right side in the form \(r^2\), the center and radius will be easier to recognize.

The first 3 terms on the left side, \(x^2-22x+121\), can be rewritten as \((x-11)^2\). The remaining terms, \(y^2+2y+1\), can be rewritten as \((y+1)^2\). The right side, 225, can be rewritten as 15^{2}. Let’s put it all together.

\((x-11)^2+(y+1)^2=15^2\)

Now we can see that the center of the circle is \((11, \text-1)\) and the circle’s radius measures 15 units.