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 x 2 = 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 ft 2 36 ft 2 32 ft 2 20 ft 2 27 ft 2 35 ft 2

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?

FigureMaximum area Equilateral triangle Square Regular Hexagon ??? Make a conjecture…. What shape would yield the most area for your dog if you have only 24 feet of fencing?