1. Solving Quadratic Equations Using the Quadratic Formula
MA.912.A.7.2 Solve quadratic equations over the real numbers by factoring and by using the quadratic formula.

2. The Quadratic Formula The solutions of a quadratic equation written in
Standard Form, ax2 + bx + c = 0, can be found by
Using the Quadratic Formula.
Have students sing song together.
Click on the link below to view a song to help you memorize it.

3. Deriving the Quadratic Formula by Completing the Square.
Divide both
sides by “a”.
Subtract constant from
both sides.

4. Deriving the Quadratic Formula by Completing the Square.
Complete
The Square
Factor the
Perfect Square
Trinomial
Simplify expression
on the left side by
finding the LCD

5. Deriving the Quadratic Formula by Completing the Square.
Take the square
root of both sides
Solve absolute value/
Simplify radical

6. Deriving the Quadratic Formula by Completing the Square.
Isolate x
Simplify
Congratulations!
You have derived
The Quadratic Formula

7. #1 Solve using the quadratic formula.
Review the meaning of the solutions as they relate to the graph of the parabola– the x intercepts of the parabola would be 2 and ⅓. Review other characteristics of the related parabola– it opens up, the vertex is the minimum, the y-intercept is 2.

8. Graph Clink on link for graphing calculator.

9. #1 Solve by factoring
Discuss with students why they might want to choose factoring instead of the quadratic formula.

10. #2 Solve by factoring This quadratic is Prime (will not factor),
Give students time to try to factor. Ask them if they can think of another way to solve since factoring in not possible– they can use the quadratic formula.
This quadratic is Prime (will not factor),
The Quadratic Formula must be used!

11. #2 Solve using the quadratic formula.
Exact
Solution
Stress to students that radicals should be left in simplest radical form. This approximate solution has been rounded to the nearest hundredth.
Ask
Approx
Solution

12. Graph Clink on link for graphing calculator.

13. #3 Solve using the quadratic formula
Discuss the fact that there are imaginary solutions for this equation, however in Algebra I we only solve over real numbers. Discuss characteristics of related parabola: “Does this parabola open up or down?”– Up, because a>0. “Does this parabola have any x-intercepts/” – No because there are no Real Solutions to the equation. “Is the vertex above or below the x-axis?”– Above, because it opens up and does not intercept the x-axis.
The is not a real number, therefore this equation has
‘NO Real Solution’

14. Graph Clink on link for graphing calculator.

15. #4 Solve using the quadratic formula
Point out the Identity property of addition (adding/subtracting zero, does not change a number). Stress to students that they should always attempt factoring before using the quadratic formula, because factoring is less time consuming and less steps. Review using the zero product property to solve and also rewrite as (x-8)2 = 0 and take the square root of both sides.
Would factoring work to solve this equation?

16. Graph Clink on link for graphing calculator.

17. #5 Solve using the quadratic formula.
Exact
Solution
Stress to students that radicals should be left in simplest radical form. This approximate solution has been rounded to the nearest hundredth.
Ask
Approx
Solution

18. #5 What if we move everything to the right side?
Exact
Solution
Stress to students that radicals should be left in simplest radical form. This approximate solution has been rounded to the nearest hundredth.
Ask
Approx
Solution

19. Graph Clink on link for graphing calculator.

20. The Discriminant
The expression inside the radical in the quadratic formula is called the Discriminant.
The discriminant can be used to determine the number of solutions that a quadratic has.

21. Understanding the discriminant
# of real solutions
Perfect
square
2 real rational solutions
Not
Perfect
2 real irrational solutions
1 real rational solution
Review definition of Rational vs. Irrational.
No real solution

22. #6 Find the discriminant and describe the solutions to the equations.
2 Real
Rational
Solutions

23. #7 Find the discriminant and describe the solutions to the equations.
No Real
Solutions

24. #8 Find the discriminant and describe the solutions to the equations.
2 Real
Rational
Solutions

25. #9 Find the discriminant and describe the solutions to the equations.
1 Real
Rational
Solution