Questions and Answers
Which of the following are steps to factoring? (Select all that apply)
What is the equation to find the vertex for the equation y=ax^2+bx+c?
x = -b/2a
How do you solve the equation 8x^2=40 by square roots?
x = rad 5
What does a complex number look like in standard form?
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What is the conjugate of the complex number 3-4i?
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What is the additive inverse of the complex number a + bi?
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Match the following operations with their results:
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To write a quadratic function in standard form with zeros -4 and 7, you need to multiply the expressions (x + ______)(x - ______).
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To evaluate two complex numbers, real parts are added and imaginary parts are added.
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Study Notes
Factoring Steps
- Rewrite the expression in standard form if necessary.
- Factor out the Greatest Common Factor (GCF) if possible.
- Utilize "magic numbers" for identifying factors.
- Apply factor by grouping for polynomials with four terms.
- Set variable to zero when factoring, e.g., for (2a = 0), (a = 0).
Solving by Graphing
- Identify the x-intercepts of the equation (y = ax^2 + bx + c).
- Find the vertex using the formula (x = -\frac{b}{2a}).
- Example: For (y = x^2 - 6x + 8), calculate (x = \frac{6}{2(1)} = 3).
- Always make a table for at least five values when graphing.
Solving by Square Roots
- Rearrange the equation to the form (x^2 = y).
- Solve by taking the square root of both sides, remembering to include the plus/minus sign.
- Example: From (8x^2 = 40) to (x^2 = 5) leads to (x = \sqrt{5}).
Factoring Techniques
- For a binomial, check for the difference of squares.
- For a trinomial, verify if it’s a perfect square trinomial and use magic numbers.
- For expressions with four terms, apply factor by grouping.
Writing Quadratic Functions
- To write a quadratic function from zeros, such as -4 and 7:
- Set each zero to zero: (x + 4 = 0) and (x - 7 = 0).
- Multiply the expressions: ((x + 4)(x - 7) = x^2 - 3x - 28).
Complex Numbers
- Imaginary unit (i) represents (\sqrt{-1}), making (i^2 = -1).
- Use (i) to express the square roots of negative numbers.
Standard Form of Complex Numbers
- The standard form of a complex number is expressed as (a + bi).
- Example: (3 + 7i) where (3) is the real part and (7i) is the imaginary part.
Evaluating Complex Numbers
- For equations like (4x + 10i = 2 - (4y)i):
- Equate real parts: (4x = 2) leading to (x = \frac{1}{2}).
- Equate imaginary parts: (10 = -4y), resulting in (y = -\frac{5}{2}).
Conjugates of Complex Numbers
- The conjugate of (a + bi) is (a - bi).
- Example: The conjugate of (3 - 4i) is (3 + 4i).
- The real part remains unchanged while the imaginary part changes sign.
Additive Inverse of Complex Numbers
- The additive inverse of (a + bi) is (-a - bi).
- Involves changing the signs for both the real and imaginary components.
Adding and Subtracting Complex Numbers
-
Addition Example:
- ((4 + 2i) + (-6 - 7i)):
- Real: (4 - 6 = -2)
- Imaginary: (2i - 7i = -5i)
- Solution: (-2 - 5i)
- ((4 + 2i) + (-6 - 7i)):
-
Subtraction Example:
- ((5 - 2i) - (-2 - 3i)):
- Real: (5 + 2 = 7)
- Imaginary: (-2i + 3i = i)
- Solution: (7 + i)
- ((5 - 2i) - (-2 - 3i)):
- Special Caution: Adding additive inverses results in zero.
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Description
This quiz covers key concepts from Algebra 2 Chapter 4, focusing on factoring and solving equations by graphing. You will learn to identify the steps involved in factoring and how to find the vertex of a quadratic equation. Enhance your understanding of these essential algebraic techniques.