A-REI Solve equations and inequalities in one variable.

1. Solve quadratic equations in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.

b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

Constant Perimeter• You have 24 feet of

fencing to make a rectangular pen for your dog. You want your dog to have as much room as possible. What are the dimensions of the pen that would give your dog the most room?

Guess and Check

10

3

6

8

6

7

5

111

2

9

4

11 ft2

36 ft2

32 ft2

20 ft2

27 ft235 ft2

Solving Algebraically

What is the meaning of the vertex in context of the problem?

What if… You have 24 feet of fencing to make a rectangular

pen for your dog. You decide to use one side of an existing fence as part of your dog’s pen. You want your dog to have as much room as possible. What are the dimensions of the pen that would give your dog the most room?

What if… You have 24 feet of fencing to make a rectangular

pen for your dog. You decide to use one side of an existing fence as part of your dog’s pen. You want your dog to have as much room as possible. What are the dimensions of the pen that would give your dog the most room?

22

Imagine the pen were in the shape of an equilateral triangle. What is the area of this triangular pen?

• Imagine the pen were in the shape of a regular hexagon. What is the area of this hexagonal pen?

Figure Maximum areaEquilateral triangle

SquareRegular Hexagon

???

Make a conjecture….

What shape would yield the most area for your dog if you have only 24 feet of fencing?