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Secondary Math 2 4.3 Complete the Square.

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Presentation on theme: "Secondary Math 2 4.3 Complete the Square."β€” Presentation transcript:

1 Secondary Math 2 4.3 Complete the Square

2 Warm Up Use the quadratic formula to solve. 3 π‘₯ 2 βˆ’13π‘₯+4=0

3 Recap of this unit so far…
We now have two different methods to solve problems in the form π‘Ž π‘₯ 2 +𝑏π‘₯+𝑐=0. We can… …factor and solve. …use the quadratic formula. There is one final method! (All three are important to know how to do.)

4 What you will learn The third method to solving quadratics: complete the square.

5 Introduction… Using any method, solve π‘₯βˆ’2 2 βˆ’25=0 π‘₯βˆ’2 π‘₯βˆ’2 βˆ’25=0 π‘₯ 2 βˆ’4π‘₯+4βˆ’25=0 π‘₯ 2 βˆ’4π‘₯βˆ’21=0 π‘₯+3 π‘₯βˆ’7 =0 π‘₯=βˆ’3, π‘₯=7

6 Introduction… Using any method, solve π‘₯βˆ’2 2 βˆ’25=0 π‘₯βˆ’2 π‘₯βˆ’2 βˆ’25=0 π‘₯ 2 βˆ’4π‘₯+4βˆ’25=0 π‘₯ 2 βˆ’4π‘₯βˆ’21=0 π‘₯= 4Β± βˆ’4 2 βˆ’4 1 βˆ’21 2(1) π‘₯=βˆ’3, π‘₯=7

7 The easy way! Solve π‘₯βˆ’2 2 βˆ’25=0 π‘₯βˆ’2 2 =25 π‘₯βˆ’2 2 = 25 π‘₯βˆ’2=Β±5 π‘₯=2Β±5 π‘₯=βˆ’3,π‘₯=7

8 Today we will learn how to write quadratics in this form.
From the introduction, we saw that π‘₯βˆ’2 2 βˆ’25=0 is the same as π‘₯ 2 βˆ’4π‘₯βˆ’21=0

9 By the end of class today, you will be able to write a quadratic function of the form π‘₯ 2 βˆ’4π‘₯βˆ’21=0 into π‘₯βˆ’2 2 βˆ’25=0.

10 Standard Form: y=π‘Ž π‘₯ 2 +𝑏π‘₯+𝑐 Vertex Form: 𝑦=π‘Ž π‘₯βˆ’β„Ž 2 +π‘˜

11 Practice some multiplication!
π‘₯+4 2 = π‘Ÿ+5 2 = π‘§βˆ’7 2 = π‘›βˆ’1 2 = Look for patterns.

12 These quadratics are called perfect squares.

13 Hint! Cut b in half, then square it. Or find 𝑏 2 2
Complete the square. π‘Ÿ 2 βˆ’12π‘Ÿ+_____ Hint! Cut b in half, then square it. Or find 𝑏 2 2 Now write it as a perfect square. Try problems 2-4.

14 4) 𝑦 2 +42𝑦+________

15 5) 𝑧 2 βˆ’7𝑧+_____ Try problems 6-8

16 6) 𝑛 𝑛+______

17 How to complete the square for quadratic equations (where a=1)
Step 1: Move the constant c to the other side of the equation. (we will do #10 as our example.) 𝑝 2 +12𝑝+27=0 𝑝 2 +12𝑝=βˆ’27

18 How to complete the square for quadratic equations (where a=1)
Step 2: Complete the square on the left side of the equation. Balance this by adding the same value to the right side of the equation. 𝑝 2 +12𝑝=βˆ’27 𝑝 2 +12𝑝+____=βˆ’27+____ 𝑝 2 +12𝑝+36=βˆ’27+36

19 How to complete the square for quadratic equations (where a=1)
Step 3: Write the left side of the equation as a perfect square. Evaluate the right side of the equation. 𝑝 2 +12𝑝+36=βˆ’27+36 𝑝+6 2 =βˆ’27+36 𝑝+6 2 =9

20 How to complete the square for quadratic equations (where a=1)
Step 4 (solving): Solve for x. You should get two solutions. Done! 𝑝+6 2 =9

21 How to complete the square for quadratic equations (where a=1)
Step 4 (vertex form): Rewrite the equation so that it equals 0. Done! 𝑝+6 2 =9 𝑝+6 2 βˆ’9=0

22 9) π‘Ž 2 +4π‘Žβˆ’15=0

23 12) π‘Ž 2 βˆ’14π‘Ž+13=0

24 13) π‘₯ 2 βˆ’3π‘₯βˆ’66=4

25 Exit Problem Write the equation in vertex form (by completing the square). π‘₯ 2 βˆ’20π‘₯+12=0


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