1. Solving Quadratic Equations by Graphing
Section 9 – 2

2. Main Ideas Solve quadratic equations by graphing.
Estimate solutions of quadratic equations by graphing.

3. Quadratic Equation: an equation of the form ax2 + bx + c = 0
Vocabulary
Quadratic Equation: an equation of the form
ax2 + bx + c = 0
,where a ≠ 0.

4. Vocabulary
The solutions of a quadratic equation are called the____________________.
The roots of a quadratic equation can be found by graphing the related quadratic function f(x) = ax2 + bx + c and finding the _____________________________.

5. d.) Graph y = x2 + 4x + 3 X Y -4 3 -3 -2 -1
Vertex 

6. f.) Solve x2 + 4x + 3 = 0 by graphing
To solve you need to know where the value of f(x) = 0. This occurs at the x-intercepts, (-3,0) and (-1,0).
The solutions are x = {-3, -1}

7. d.) Graph x2 + 7x + 12 = 0 X Y -5 2 -4
-3.5
-0.25 -3 -2
Vertex 

8. f.) Solve x2 + 7x + 12 = 0 by graphing
To solve you need to know where the value of f(x) = 0. This occurs at the x-intercepts, (-3,0) and (-4,0).
The solutions are x = {-4, -3}

9. d.) Graph x2 – 4x + 5 = 0 X Y 5 1 2 3 4
Vertex 

10. f.) Solve x2 – 4x + 5 = 0 by graphing
To solve you need to know where the value of f(x) = 0. There is no x- intercepts, so there is no real roots
Solution: {Ø}

11. d.) Graph x2 – 6x + 9 = 0 X Y 1 4 2 3 5
Vertex 

12. f.) Solve x2 – 6x + 9 = 0 by graphing
To solve you need to know where the value of f(x) = 0. This occurs at the x-intercepts (3,0).
Solution: {3}

13. Solving Quadratic Equations by Graphing
Section 9 – 2
Day 2

14. Main Ideas Solve quadratic equations by graphing.
Estimate solutions of quadratic equations by graphing.

15. Estimate Solutions
The roots of a quadratic equation may not be integers. If exact roots cannot be found, they can be estimated by finding the consecutive integers between which the roots lie.

16. Example 2: Solve the equation by graphing
x2 + 6x + 6 = 0 a.) Write the equation of the axis of symmetry b.) Find the coordinates of the vertex of the graph of each function. c.) Identify the vertex as a maximum or a minimum. d.) Then graph the function. e.) State the Domain and Range f.) Determine what the x-intercept or zeros are to solve the quadratic equation.

17. Solve x2 + 6x + 6 = 0 by graphing
a.) equation of axis of symmetry a = _____ b = ____ c = ____ b.) Find the vertex. 1 6 6
y = x2 + 6x + 6 when x = -3
y = (-3)2 + 6(-3) + 6
y =
y = -3
Vertex (-3, -3)

18. Solve x2 + 6x + 6 = 0 by graphing
c.) Identify the vertex as a maximum or a minimum. Since the coefficient of the x2-term is positive, the parabola opens upward, and the vertex is a minimum point.

19. d.) Graph x2 + 6x + 6 = 0 X Y -5 1 -4 -2 -3 -1
Vertex 

20. Domain: D: {x I x is all real numbers.} Range: R: {y I y ≥ -3}
e.) Graph x2 + 6x + 6 = 0
Domain: D: {x I x is all real numbers.} Range: R: {y I y ≥ -3}

21. f.) Solve x2 + 6x + 6 = 0 by graphing
The x-intercepts of the graph are between -5 and -4 and between -2 and -1.
So one root is between 5 and 4, and the other root is between 2 and 1.
-5 < x < -4
-2 < x < -1

22. Try 4: Solve the equation by graphing
x2 – 2x – 4 = 0 a.) Write the equation of the axis of symmetry b.) Find the coordinates of the vertex of the graph of each function. c.) Identify the vertex as a maximum or a minimum. d.) Then graph the function. e.) Determine what the x-intercept or zeros are to solve the quadratic equation.

23. Solve x2 – 2x – 4 = 0 by graphing
a.) equation of axis of symmetry a = ____ b = ____ c = ____ b.) Find the vertex. 1 -2 -4
y = x2 – 2x – 4 when x = 1
y = (1)2 + -2(1) + -4
y =
y = -5
Vertex (1, -5)

24. Solve x2 – 2x – 4 = 0 by graphing
c.) Identify the vertex as a maximum or a minimum. Since the coefficient of the x2-term is positive, the parabola opens upward, and the vertex is a minimum point.

25. d.) Graph x2 – 2x – 4 = 0 X Y -1 -4 1 -5 2 3
Vertex 

26. e.) Domain and Range
Domain: D: {x I x is all real numbers.} Range: R: {y I y ≥ -5}

27. Solve x2 + 6x + 6 = 0 by graphing
The x-intercepts of the graph are between -2 and -1 and between 3 and 4.
So one root is between - 2 and -1, and the other root is between 3 and 4.
-2 < x < -1
3 < x < 4

28. 2 roots x = { -4, 3} x – 3 = 0 x + 4 = 0 x + -3 + 3 = 0 + 3
Use factoring to determine how many times the graph of each function intersects the x-axis. Identify the root.
f(x) = x2 + x – 12 0 = x2 + x – 12 0 = (x – 3)(x + 4)
x – 3 = 0
x = 0 + 3
x = 3
x + 4 = 0
x =
x = - 4
2 roots x = { -4, 3}