Transcript 2.1.6 solve quadratic equation notes
2.1.6 Quadratic Equations
Essential Question: How do you solve quadratic equations and what do the solutions represent?
Main Idea
• • Today, we are going to discuss and review how to solve quadratic equations. You have learned about this in Algebra I.
•
Today’s methods for solving quadratic equations:
–
Graphing
–
Square Roots
–
Factoring
Solving
•
To solve a quadratic function, look for the x- intercepts: where the parabola intersects the x-axis.
• • • • •
Words for solutions to quadratics: 1) zeros 2) roots 3) solutions 4) answers 5) x-intercepts
Solutions
• • A zero is just like looking for an x-intercept A quadratic will only have
2, 1, or no zeros
.
Solving by Graphing
1) x
2 - 9 = 0
Practice
Main Idea
• Some quadratic equations are missing an a, b, or c value, and graphing can be difficult.
• • •
You can solve by square roots.
Remember, you cannot take the square root of a negative number. However, there are 2 answers to a square root: the positive and negative root.
Squares & Square Roots
• Squares and square roots are inverse operations because they are opposites.
Finding Squares
• Squares up to 30.
1 2 = 2 2 =
1 4
3 2 = 4 2 = 5 6 2 2 = =
9 16 25 36
7 2 = 8 2 =
49 64
9 2 =
81
10 2
= 100
11 2
= 121
12 2
= 144
13 2
= 169
14 2
= 196
15 2
= 225
16 2
= 256
17 2
= 289
18 2
= 324
19 2
= 361
20 2
= 400
21 2 = 22 2 = 23 2 = 24 2 = 25 2 = 26 2 = 27 2 = 28 2 = 29 2 = 30 2 =
441 484 529 576 625 676 729 784 841 900
Solving by Square Roots
•
To solve by square root:
1) Quadratic does NOT have a b. Only ax
2 + c = 0
2) Isolate the variable by inverse operations.
3) Square root both sides, if possible.
4) Write solution as + and – answer.
Simplify, then Root
1) Isolate x 2 2) Take the square root of both sides
Approximating Solutions
• When factoring doesn’t work, and it isn’t a perfect square, then you must round your answer or simplify the radical.
Predicting with Evidence
• What are the zeros for y = x 2 – x – 6? • • Now factor x
2
solutions?
– x – 6 = 0
and solve. What do you notice about the graph and our factored What does this mean?
Zero Product Property
• • • Zero Product Property: if the product equals zero, then one of the values must equal zero. This property helps to find our ZEROS.
Meaning: Our quadratic equation must equal 0 in order to find the zeros by solving.
Example 2: Solving by Factoring
•
To Solve by factoring:
1) Write the quadratic equation in standard form: y = ax
2 + bx + c OR ax 2 + bx + c = 0
2) Factor the quadratic equation (2.1.5) 3) Use the Zero Product Property to set each factor to 0 and solve.
The answers are your zeros on the graph!
Ex 2- Solving Quadratic Equations By Factoring
Ex 2- Solving Quadratic Equations By Factoring
Ex 3 Sports Application Same steps: 1) Write in standard form 2) 3) Factor Solve each factor to find the zeros!
Practice 3
Summary
• Answer the essential question in detailed, complete sentences.
•
How do you solve quadratic equations and what do the solutions represent?
• Write 3-5 study questions in the left column to correspond with the notes.