1. Quadratics Learning Goals:
Review
Quadratics
Learning Goals:
I can put a standard form quadratic equation into vertex form by completing the square
I can answer word problems involving quadratics

2. Review
Quadratics
Quadratics that are perfect squares have “c” values that are equal to ( 𝑏 2 )2.
For example: y = x2 + 4x + 4 is a perfect square trinomial because:
b = 4
c = ( 4 2 )2
= 22
= 4

3. x x2 1 Quadratics Why is y = x2 + 4x + 4 called a perfect square?
Review
Quadratics
Why is y = x2 + 4x + 4 called a perfect square?
Algebra Tiles x2 x 1

4. Review
Quadratics
We can represent y = x2 + 4x + 4 using algebra tiles: x 1 1 x 1 1 x x x2

5. a2x2 ±2abx + b2 (ax ± 𝑏 ) 2 Quadratics
Review
Quadratics
Perfect Square Trinomial are always in this form:
a2x2 ±2abx + b2
And when we factor a perfect square trinomial, it will always factor like this:
(ax ± 𝑏 ) 2

6. Review
Quadratics
Is this an example of a perfect square trinomial?
16x2 +8x +2

7. Review
Quadratics
Factor this perfect square trinomial:
25x2 - 30x + 9

8. Review
Quadratics
Remember that you can factor out any number, even if it leaves you with a rational number.
Example: y = 2x2 + 3x + 9

9. Review
Quadratics
Determine the “c” value that would make each of the following perfect squares:
a) y = x2 + 5x + c

10. Review
Quadratics
Determine the “c” value that would make each of the following perfect squares:
c) y = x2 + 9x + c

11. Review
Quadratics
Completing the square is the process we go through to change a standard form quadratic equation into vertex form.

12. Standard Form Vertex Form
Review
Quadratics
What do you notice?
Standard Form
Vertex Form
x2 + 2x + 3
(x + 1)2 + 2
x2 + 4x + 1
(x + 2)2 - 3
x2 + 6x + 8
(x + 3)2 – 1
x2+ 4x + 5
2(x + 1)2 + 3
3x2 + 18x + 12
3(x + 3)2 - 15
2x2 + 8x + 7
2(x + 2)2 - 1

13. Quadratics Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Review
Quadratics
Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Step One: If there is an “a” value, factor it out of the ”x2” and the “x” term.

14. Quadratics Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Review
Quadratics
Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Step Two: Find a constant that must be added inside the brackets to make a perfect square.

15. Quadratics Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Review
Quadratics
Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Step Three: Inside the brackets, add and subtract the constant you found in step 2.

16. Quadratics Example1 : Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Review
Quadratics
Example1 : Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Step Four: Group the three terms that make the perfect square together. Move the subtracted term outside the brackets by first multiplying it by the common factor.

17. Quadratics Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Review
Quadratics
Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Step Five: Factor the perfect square.

18. Quadratics Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Review
Quadratics
Example 1: Put 𝑦 = 𝑥2 – 12𝑥 + 5 into vertex form
Step Six: Collect like terms.

19. Quadratics Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Review
Quadratics
Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Step One: If there is an “a” value, factor it out of the ”x2” and the “x” term.

20. Quadratics Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Review
Quadratics
Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Step Two: Find a constant that must be added inside the brackets to make a perfect square.

21. Quadratics Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Review
Quadratics
Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Step Three: Inside the brackets, add and subtract the constant you found in step 2.

22. Quadratics Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Review
Quadratics
Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Step Four: Group the three terms that make the perfect square together. Move the subtracted term outside the brackets by first multiplying it by the common factor.

23. Quadratics Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Review
Quadratics
Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Step Five: Factor the perfect square.

24. Quadratics Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Review
Quadratics
Example 2: Put 𝑦 = 5𝑥2 + 15𝑥 − 7 into vertex form
Step Six: Collect like terms.

25. Review
Quadratics
Example 3: Put 𝑦 = 1 2 𝑥2 −3𝑥+5 into vertex form

26. Review
Quadratics
A football is kicked into the air. Its height, ℎ, in metres, after 𝑡 seconds is approximated by the equation: ℎ = −5(𝑡 – 2.3 ) What is the maximum height reached by the football?

27. Review
Quadratics
A computer game company models the profit on its latest game using the equation : P = −3 𝑥 – , where 𝑥 is the number of games sold in hundred thousands, and 𝑃 is the profit in millions of dollars. What is the maximum profit the company can expect to earn using this model, and how many games do they need to sell to make that profit?

28. Diagnostic Test - Tuesday
Review
Quadratics
Homework
Pg. 2 #8-10, 13
Diagnostic Test - Tuesday