1. Prepared by Doron Shahar
Chapter 1 Section 1.4
Quadratic Equations

2. Warm-up: page 15 The zero product property says that if ,
Prepared by Doron Shahar
Warm-up: page 15
A quadratic equation is an equation that can be written in the form _____________ where a, b, and c are constants and a ≠ 0.
The zero product property says that if ,
then either ________ or ________.

3. FOIL and Factoring Factor FOIL First Outside Inside Last
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FOIL and Factoring
FOIL
Factor
First
Outside
Inside
Last

4. 1.4.1 Solve by Factoring Starting Equation Factor
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1.4.1 Solve by Factoring
Starting Equation
Factor
Zero Product Property
Solution

5. 1.4.2 Solve by Factoring Starting Equation Factor
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1.4.2 Solve by Factoring
Starting Equation
Factor
Zero Product Property
Solution

6. Solve by Factoring Starting Equation FOIL Factor Zero Product Property
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Solve by Factoring
Starting Equation
FOIL
Factor
Zero Product Property
Solution

7. Intro to Completing the square
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Intro to Completing the square
Starting Equation
Take square root of both sides of the equation
Place ± on the right side of the equation
Solution

8. Intro to Completing the square
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Intro to Completing the square
Starting Equation
Take square root
Insert ± on right side
Group like terms
Solution

9. Goal of Completing the square
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Goal of Completing the square
The goal of completing the squares is to get a quadratic equation into the following form:
Starting Equation
eg.
Take square root
Insert ± on right side
Group like terms
Solution

10. 1.4.5 Completing the square Starting Equation Add 3 to both sides
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1.4.5 Completing the square
The goal of completing the squares is to get a quadratic equation into the following form:
Starting Equation
Add 3 to both sides
Multiply both sides by 2
Desired form

11. Example: Completing the square
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Example: Completing the square
Starting Equation
Add 8 to both sides
Multiply both sides by 2
Desired form

12. Completing the square Equation from previous slide Take square root
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Completing the square
Equation from previous slide
Take square root
Insert ± on right side
Group like terms
Solution

13. 1.4.2 General method of Completing the square
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1.4.2 General method of Completing the square
Starting Equation
Add 9 to both sides
Add (−8/2)2=16
to both sides
Factor left side

14. 1.4.4 General method of Completing the square
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1.4.4 General method of Completing the square
Starting Equation
Divide both sides by 3
Add ((−2/3)/2)2=1/9
to both sides
Factor left side

15. 1.4.3 General method of Completing the square
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1.4.3 General method of Completing the square
Starting Equation
Subtract 22 from both sides
Add (−6/2)2=9
to both sides
Factor left side

16. No solutions in quadratic equation
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No solutions in quadratic equation
Try solving
Take square root
¡PROBLEMA! You CANNOT take the square root of a
negative number. Therefore, the equation has no solution.
Solving by factoring works only if the equation has a solution.
Completing the square always works, and can be used to
determine whether a quadratic equation has a solution.

17. General method of Completing the square
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General method of Completing the square
Starting Equation
Subtract c from both sides
Divide both sides by a
Add ((b/a)/2)2=(b/2a)2
to both sides
Factor left side

18. Quadratic Formula Quadratic Formula Starting Equation Solution
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Quadratic Formula
If we solve for in the previous equation, ,
we get an equation called the quadratic formula.
Quadratic Formula
The quadratic formula gives us the solutions to every quadratic equation.
Starting Equation
Solution

19. Quadratic Formula Song
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Quadratic Formula Song
Quadratic Formula
Please sing along.

20. Using the quadratic formula
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Using the quadratic formula
Starting Equation
Plug 9 in for a, 6 for b, and 1 for c in the quadratic formula.
Quadratic Formula
Solution

21. Simplify your solution
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Simplify your solution
Simplify
Solution

22. 1.4.1 Using the quadratic formula
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1.4.1 Using the quadratic formula 1
Starting Equation
Plug 1 in for a, 9 for b, and 14 for c in the quadratic formula.
Quadratic Formula
Solution

23. Simplify your solution
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Simplify your solution
Simplify
Solution

24. 1.4.3 Using the quadratic formula
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1.4.3 Using the quadratic formula
+( )
Starting Equation 1
Plug 1 in for a, −6 for b, and 22 for c in the quadratic formula.
Quadratic Formula
Solution

25. Simplify your solution
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Simplify your solution
Simplify
You cannot take the square root of a negative number. Therefore, there is no solution.
No Solution

26. Discriminant Equation Discriminant # of Solutions
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Discriminant
Equation
Discriminant
# of Solutions

27. Prepared by Doron Shahar
Calculator
Put the Quadratic Formula program on your calculator.
Instructions are in the back of the class notes. OR
You can come in to office hours to have me load the program onto your calculator.
Warning!
The calculator will not always give you exact answers.

28. Simplifying expressions with
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Simplifying expressions with

29. Quadratic equations with Decimals
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Quadratic equations with Decimals
If a quadratic equation has decimals, it is easiest to simply use the quadratic formula. If you want, you can multiply both sides of the equation by a power of 10 (i.e., 10, 100, 1000, etc) to get rid of the decimals. This can make it easier to simplify the answer if you are evaluating the quadratic formula without a calculator.

30. Quadratic equations with Fractions
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Quadratic equations with Fractions
If a quadratic equation has fractions (and does not factor), it is often easy to simply use the quadratic formula. If you want, you can multiply both sides of the equation by the least common denominator to get rid of the fractions. This can make it easier to simplify the answer.
If a quadratic equation has fractions (and factors), it is often easier to factor after having gotten rid of the fractions.

31. Variables in the denominators
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Variables in the denominators
If an equation has variables in the denominator, it is NOT a quadratic equation. Such equations, however, can lead to linear equations.
We treat such equations like those with fractions. That is, we multiply both sides of the equation by a common denominator to get rid of the variables in the dominators. Ideally, we should multiply by the least common denominator.
Example:
If our problem has B +16 in the denominator of one term, and B in the denominator of another term, we multiply both sides of the equation by B(B+16). After the multiplication, the terms will have no variables in the denominators.

32. Variables in the denominators
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Variables in the denominators
Starting Equation
Multiply both sides by B(B+16)
Distribute the B(B+16)
The problem is now in a form you can solve.

33. Prepared by Doron Shahar
Systems of equations
There are two methods for solving systems of equations: Substitution and Elimination. Both work by combining the equations into a single equation with one variable. And sometimes the resulting equation leads to a quadratic equation.
We will only review substitution, because elimination is not a common method when working with systems of equations that lead to quadratic equations.

34. Substitution Starting Equations Substitute B+16 for J
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Substitution
Starting Equations
Substitute B+16 for J
in the second equation
Solution for B
Plug in 24 and −10 for B in the first equation to get the solution for J