1. Solving Quadratic Equations Using the Quadratic Formula
11.2
Solving Quadratic Equations Using the Quadratic Formula
1. Solve quadratic equations using the quadratic formula.
2. Use the discriminant to determine the number of real solutions that a quadratic equation has.
3. Find the x- and y-intercepts of a quadratic function.
4. Solve applications using the quadratic formula.

2. Coefficient of squared term is NOT 1.
Solve:
Coefficient of squared term is NOT 1.
squared
Quadratic Formula

3. Quadratic Formula
To solve ax2 + bx + c = 0, where a 0, use

4. 2 real rational solutions
Solve:
a = 2
, b = –7
, c = 3
2 real rational solutions

5. 2 real irrational solutions
Solve:
a = 1, b = –2, c = –11. 2 1 1
16 ∙3
2 real irrational solutions

6. 2 non-real complex solutions
Solve:
a = 1, b = -4, c = 5 2 1 1
2 non-real complex solutions

7. Copyright © 2011 Pearson Education, Inc.
Solve using the quadratic formula. a) b) c) d)
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11.2

8. Copyright © 2011 Pearson Education, Inc.
Solve using the quadratic formula. a) b) c) d)
Copyright © 2011 Pearson Education, Inc.
11.2

9. Methods for Solving Quadratic Equations
When the Method is Beneficial
1. Factoring
Use when the quadratic equation can be easily factored.
2. Square root principle
Use when the quadratic equation can be easily written in the form
No middle term.
3. Completing the square
Rarely the best method, but important for other topics.
4. Quadratic formula
Use when factoring is not easy, or possible.

10. What made the difference? The Discriminant
2 non-real
complex solutions
2 real rational
solutions
2 real irrational
solutions
What made the difference?
The Discriminant

11. Discriminant:
The discriminant is the radicand, b2 – 4ac, in the quadratic formula.
The discriminant is used to determine the number and type of solutions to a quadratic equation.

12. 2 non-real complex solutions 2 real rational solutions
2 real irrational
solutions
If the discriminant is….
there will be….
positive and a perfect square
2 real rational solutions. There will be no radicals left in the answer. The equation could have been factored.
positive but not a perfect square
2 real irrational solutions. There will be a radical in the answer.
1 real rational solution.
negative
2 non-real complex solutions. The answer will contain an imaginary number.

13. Evaluate the discriminant: b2 – 4ac.
Use the discriminant to determine the number and type of solutions.
Evaluate the discriminant: b2 – 4ac.
a = 2, b = 5, c = 1
Discriminant:
Positive but not a perfect square.
Two real irrational solutions.

14. Copyright © 2011 Pearson Education, Inc.
Find the discriminant.
a) 5
b) 73
c) 25 d)
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15. Copyright © 2011 Pearson Education, Inc.
Find the discriminant.
a) 5
b) 73
c) 25 d)
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11.2

16. Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
Copyright © 2011 Pearson Education, Inc.
11.2

17. Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
Copyright © 2011 Pearson Education, Inc.
11.2

18. Copyright © 2011 Pearson Education, Inc.
Find the discriminant. a)
b) 136 c) d)
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11.2

19. Copyright © 2011 Pearson Education, Inc.
Find the discriminant. a)
b) 136 c) d)
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11.2

20. Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
Copyright © 2011 Pearson Education, Inc.
11.2

21. Copyright © 2011 Pearson Education, Inc.
Determine the number and type of solutions.
a) Two real rational solutions
b) Two real irrational solutions
c) One real rational solution.
d) Two non-real complex solutions.
Copyright © 2011 Pearson Education, Inc.
11.2

23. What are we finding?
x-intercepts
y-intercept