1. Solving Quadratic Equations

2. Solving Quadratic Equations
What is the definition of a solution to a quadratic equation?
A solution is the value of x when y = 0.
What are other terms for the solutions to a quadratic equation?
roots
zeros
x-intercepts
(in some cases)

3. Graphs and Solutions to Quadratic Functions: 3 Cases
Two x-intercepts = two real solutions (rational or irrational)
One x-intercept = one real solution (always rational)
No x-intercepts = two complex solutions
ALL QUADRATIC EQUATIONS CAN BE SOLVED!

4. Name the 4 methods of solving quadratics
factoring
The Square Root
The Quadratic Formula
Completing the Square

5. Method 1: Solving Quadratic Equations by Factoringok ok
need this to be 0
Let's solve the equation
1. First you need to get it in what we call “standard form" which means
Factoring is the easiest way to solve a quadratic equations, but it won’t work for all functions, as many cannot be factored!
Subtract 18
2. Now let's factor the left hand side
3. Now set each factor = 0 and solve for each answer.
Meaning: 2 x-intercepts, 2 real solutions

6. Method 2: The Square Root: ax2 + c = 0
This method will work for any equation that doesn’t have a “bx” term, it only has “ax2” or “a(x-h)2” and a constant. The objective is to get x2 alone on one side of the equation and then take the square root of each side to cancel out the square.
1. Get the "squared stuff" alone which in this case is the t 2 25 5 5
2. Now square root each side.
Don’t forget that (-5)(-5) = 25 also!
Meaning: 2 x-intercepts, 2 real solutions

7. Let's try another one:
1. Get the "squared stuff" alone which in this case is the u 2 4 4
2. Square root each side.
Remember with a fraction you can square root the top and square root the bottom
DON'T FORGET BOTH THE + AND –
Hey, what about the – under the square root?
Recall , so x equals two imaginary numbers!
Meaning: no x-intercepts, but there are still 2 solutions.

8. You try

9. Another Example: “a(x-h)2”
1, Get the "squared stuff" alone (i.e, the parentheses)
2. Now square root each side and DON'T FORGET BOTH THE + AND –
25 · 2
Let's simplify the radical -1 -1
Now solve for x
Meaning: two x-intercepts, but they are irrational.

10. Perfect Square Trinomials: What’s the pattern?
Add how much?
c = ?
Factored form 1 4 9 16
“add half of b squared”
To complete the square and make a perfect square trinomial,_________________

11. What completes the square?
100 36
81/4
You can only complete the square when a = 1!
No Solution

12. Method 4: The Quadratic Formula
The Quadratic Formula is a formula that can solve any quadratic, but it is best used for equations that cannot be factored or when completing the square requires the use of fractions. It is the most complicated method of the four methods.
Do you want to see where the formula comes from?

13. The Quadratic Formula 5. Simplify radical
This formula comes from completing the square of a quadratic written in standard form
5. Simplify radical
1. Subtract c and Divide by a
6. Get x alone
2. Complete the square:
7. Simplify right hand side
3. Factor left side, combine right side
4. Square root each side

14. “x equals opposite b plus or minus square root of b squared minus 4ac all over 2a”

15. two complex solutions (no x-intercepts)
This part of the formula is called the “Discriminant”
The discriminant tells us what kind of solutions we have:
One real solution one x -intercept (always rational)
two complex solutions (no x-intercepts)
Two real solutions two x-intercepts (rational or irrational)

16. The Quadratic Formula (4) 1. Identify a, b, c 6. Simplify a = 4 b= 2
Solve the equation
Notice the solutions are complex!
1. Identify a, b, c
6. Simplify
a = 4
b= 2
c = 5
=4•19
2. Plug into the formula
7. Simplify radical 2
(2)2
(4)
(5)
(4)
8. Simplify final answer, if possible
5. Simplify
Meaning: 0 x-intercepts, 2 complex solutions

17. The Quadratic Formula ( ) 1. Identify a, b, c 6. Simplify a = b= c =
Solve the equation
1. Identify a, b, c
6. Simplify
a = b=
c =
2. Plug into the formula
7. Simplify radical
( )2
( )
( )
( )
8. Simplify final answer, if possible
5. Simplify

18. The Quadratic Formula ( 1 ) 1. Identify a, b, c 6. Simplify a = 1 b= 1
Solve the equation
1. Identify a, b, c
6. Simplify
a = 1
b= 1
c = -1
2. Plug into the formula
7. Simplify radical
Already simplified 1
( 1)2
( 1 )
(-1 )
( 1 )
8. Simplify final answer, if possible
5. Simplify

19. Another example
Solve the equation

20. “No solution” could never be an answer to a quadratic equation.
True or False?
All quadratic equations have solutions.
“No solution” could never be an answer to a quadratic equation.
TRUE.
You can solve ANY quadratic equation, you just may need to use a particular method to get to the answers.

21. All quadratics have solutions. It is the value of x when y = 0.
True or False?
The solutions (zero, root) to a quadratic equation are always x-intercepts on its graph.
False
All quadratics have solutions. It is the value of x when y = 0.
But not all quadratics cross the x – axis, so the solutions will not always be x-intercepts.

22. Neither of these functions have x-intercepts, but they still have two complex solutions

23. True or False? False All quadratics equations can be factored.
Here are just a few examples of quadratics that cannot be factored:

24. False. You will need to have the formula memorized.
True or False?
The quadratic formula will be provided to you for the test and final exam.
False. You will need to have the formula memorized.

25. Which method should you use?
Solve
(x+1)(x+5) b. x = -1, x = c. x = 1, x = d. no sol
Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?
The Square Root
factoring
The Quadratic Formula

26. Which method should you use?
Solve b.
c. no sol because you cannot factor it
Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?
The Square Root
factoring
The Quadratic Formula

27. Which method should you use?
Solve b.
c. no sol because you cannot factor it
Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?
The Square Root
factoring
The Quadratic Formula

28. Which method should you use?
Solve b.
c d.
Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?
The Square Root
factoring
The Quadratic Formula

29. Which method should you use?
a b c.
Stop! Before you begin to solve this problem, look at the possible solutions. What method should you use to solve this problem?
The Square Root
factoring
The Quadratic Formula

30. Completing the Square: Use #1
This method is used for quadratics that do not factor, although it can be used to solve any kind of quadratic function.
1. Get the x2 and x term on one side and the constant term on the other side of the equation.
2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this.
3. Factor the trinomial
4. Apply the square root and solve for x

31. Completing the Square: Use #2
By completing the square, we can take any equation in standard form and find its equation in vertex form: y = a(x-h)2 + k
1. Get the x2 and x term on one side and the constant term on the other side of the equation.
2. To “complete the square,”, add “half of b squared” to each side. You will make a perfect square trinomial when you do this.
What is the vertex?
3. Factor the trinomial.
4. Write in standard form.