1. Solving Quadratic Equation by Graphing and Factoring
Section 6.2& 6.3
CCSS: A.REI.4b

2. Mathematical Practices:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.  
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning. 

3. CCSS: A.REI.4b
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.

4. Essential Question:
How do I determine the domain, range, maximum, minimum, roots, and y-intercept of a quadratic function from its graph & how do I solve quadratic functions by factoring?

5. Quadratic Equation y = ax2 + bx + c ax2__ is the quadratic term.
bx--- is the linear term.
c-- is the constant term.
The highest exponent is two; therefore, the degree is two.

6. Identifying Terms Example f(x)=5x2-7x+1 Quadratic term 5x2
Linear term x
Constant term 1

7. Identifying Terms Example f(x) = 4x2 - 3 Quadratic term 4x2
Linear term
Constant term

8. Identifying Terms Now you try this problem. f(x) = 5x2 - 2x + 3
quadratic term
linear term
constant term
5x2
-2x 3

9. Quadratic Solutions The number of real solutions is at most two.
No solutions
One solution
Two solutions

10. Solving Equations
When we talk about solving these equations, we want to find the value of x when y = 0. These values, where the graph crosses the x-axis, are called the x-intercepts.
These values are also referred to as solutions, zeros, or roots.

11. Identifying Solutions
Example f(x) = x2 - 4
Solutions are -2 and 2.

12. Identifying Solutions
Now you try this problem.
f(x) = 2x - x2
Solutions are 0 and 2.

13. Graphing Quadratic Equations
The graph of a quadratic equation is a parabola.
The roots or zeros are the x-intercepts.
The vertex is the maximum or minimum point.
All parabolas have an axis of symmetry.

14. Graphing Quadratic Equations
One method of graphing uses a table with arbitrary
x-values.
Graph y = x2 - 4x
Roots 0 and 4 , Vertex (2, -4) ,
Axis of Symmetry x = 2 x y 1 -3 2 -4 3 4

15. Graphing Quadratic Equations
Try this problem y = x2 - 2x - 8.
Roots
Vertex
Axis of Symmetry x y -2 -1 1 3 4

16. Graphing Quadratic Equations
The graphing calculator is also a helpful tool for graphing quadratic equations.

17. Roots or Zeros of the Quadratic Equation
The Roots or Zeros of the Quadratic Equation are the points where the graph hits the x axis.
The zeros of the functions are the input that make the equation equal zero.
Roots are 4,-3

18. To solve a Quadratic Equation
Make one side zero.
Then factor then set each factor to zero

19. Solve

20. Solve

21. Solve

22. Solve

23. Solve

24. Solve
Solve

25. Solve
Multiply the ends together and find what adds to the coefficient of the middle term

26. Solve
Use -6 and 1 to break up the middle term

27. Solve
Use group factoring to factor, first two terms and then the last two terms

28. Solve

29. How to write a quadratic equation with roots
Given r1,r2 the equation is (x - r1)(x - r2)=0
Then foil the factors,
x2 - (r1 + r2)x+(r1· r2)=0

30. How to write a quadratic equation with roots
Given r1,r2 the equation is (x - r1)(x - r2)=0
Then foil the factors,
x2 - (r1 + r2)x+(r1· r2)=0
Roots are -2, 5
Equation x2 - (-2+5)x+(-2)(5)=0
x2 - 3x -10 = 0

31. How to write a quadratic equation with roots
Roots are ¼, 8
Equation x2 -(¼+8)x+(¼)(8)=0
x2 -(33/4)x + 2 = 0
Must get rid of the fraction, multiply by the common dominator. 4
4x2 - 33x + 8 = 0