Warmups A. B. C. D.

3-5 Completing the Square Goal: Solve quadratic equations by using the Square Root Property. Solve quadratic equations by completing the square.

Square Root Property For any real number n, if 𝑥 2 = n, then 𝑥= ± 𝑛 This works for blob math too! If (𝑏𝑙𝑜𝑏) 2 = n, then (𝑏𝑙𝑜𝑏)= ± 𝑛 . Blob can be any kind of expression. All quadratic equations can be re-written as an expression squared equal to a number.

For example, if asked to solve the equation using the Square Root Property, you could create a perfect square expression on the left involving the x-values, and on the right just have a number. 𝑥 2 +10𝑥+25=7

Solve the equation by completing the square. 𝑥 2 +4𝑥+11=0 If there is a lead coefficient, you MUST divide out the lead coefficient first. You can only complete the square with a lead coefficient of one.

In previous sections we graphed quadratic functions and talked a little about transformations – reflecting them to open downward, moving them left and right, stretching and shrinking them. How do you write a quadratic function in the form of y= 𝑓 𝑥 =𝑎 𝑥 2 +𝑏𝑥+𝑐 so you can find the vertex easily, and see the transformation easily? You use the completing the square process! The new format looks like y=𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘

y=𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘 Variable Effects on graph a (lead coefficient) 𝑎>0, opens upward 𝑎<0, opens downward 𝑎 >1, stretch by a factor of 𝑎 𝑎 <1, shrink by a factor of 𝑎 h Horizontal translation (direction depends on what h-value makes expression zero) Inside the house k Vertical translation (what you see is what you get) Outside the house

2. 𝑓 𝑥 = 𝑥 2 +3𝑥 −10 y-int Max or min Vertex form Vertex Line of Symm Value of max or min x-int’s

6. 𝑓 𝑥 = 3𝑥 2 −24𝑥+50 y-int Max or min Vertex form Vertex Line of Symm Value of max or min x-int’s

14. An orange grower finds that she gets an average yield of 40 bushels per tree when she plants 20 trees on an acre of ground. Each time she adds a tree to an acre, the yield per tree decreases by one bushel, due to crowding. How many trees per acre should she plant for maximum yield? What is the maximum yield? (Two answers)