Questions and Answers
What is the process to find the absolute value of each complex number?
Use a^2 + b^2 = c^2 and solve for the square root.
How do you simplify equations that are divided?
Take the bottom equation and multiply it on both top and bottom. Distribute everything.
How do you graph each number in the complex plane?
Real number first, then imaginary. Real = x, imaginary = y. Set it as a plotting point.
What is the method to simplify an equation straight across?
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What is the value of i^2?
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How do you find the discriminant value of a quadratic equation and determine the type of solutions?
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How do you write a polynomial function of least degree with integral coefficients given the zeros?
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What is the process to sketch the graph of an inequality?
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How do you graph each inequality?
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How do you solve each inequality and graph its solution on the number line when it's greater than?
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How do you solve each inequality and graph its solution on the number line when it's less than?
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What is the method to solve each quadratic inequality and graph the solutions on the number line?
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How do you solve each equation by factoring?
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What is the process to solve each equation by taking square roots?
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How do you solve each equation using the quadratic formula?
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What steps do you take to sketch the graph of each function and identify the roots/zeros if possible?
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What is the process to simplify square roots?
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What should you do if the equation has square roots on both ends?
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Study Notes
Absolute Value of Complex Numbers
- Calculate using the formula a² + b² = c² and find the square root.
Simplifying Divided Equations
- Multiply the denominator by both the numerator and denominator; distribute thoroughly.
Graphing in the Complex Plane
- Plot the real part as x and the imaginary part as y on the complex plane.
Simplifying Equations
- Combine like terms; for multiplication, perform operations from left to right.
Value of i²
- The value of i² is defined as -1.
Discriminant of Quadratic Equations
- Use b² - 4ac to determine nature of solutions:
- Negative root: complex solutions
- Zero: one solution
- Positive integer root: real solutions
Writing Polynomial Functions from Zeros
- Use complex conjugates and multiply each pair to form a polynomial function with integral coefficients.
Graphing Inequalities
- Use a calculator for skirting the inequality; a dotted line indicates “not equal to,” and parabolic shapes signify the function type.
Graphing Each Inequality
- Enter the inequality into a calculator; the resulting graph forms a "V" shape, with dotted lines for non-inclusive inequalities.
Solving and Graphing "Greater Than" Inequalities
- Break into two equations; flip the inequality for negative solutions; solve algebraically.
Solving and Graphing "Less Than" Inequalities
- Solutions should be both negative and positive; achieve m isolation; note signs only flip when dividing by a negative.
Quadratic Inequalities Solution Method
- Set equal to zero, factor, and use the Zero Product Property (ZPP) to find solutions.
Factoring to Solve Equations
- Substitute values to check if the equation holds true after factoring.
Solving Equations via Square Roots
- Isolate x on one side and take the square root of both sides.
Quadratic Formula for Equation Solutions
- Employ the formula x = -b ± √(b² - 4ac) / 2a for solving quadratics.
Graphing Functions and Identifying Roots
- After graphing, determine roots:
- No intersection with x-axis: no solutions
- Touching once: one solution
- Touching twice: two solutions; identify the roots.
Simplifying Square Roots
- Extract perfect squares from under the radical and represent them separately outside the square root.
Handling Equations with Square Roots on Both Sides
- Square both sides of the equation to eliminate square roots, then proceed to solve.
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Description
This quiz covers various concepts related to complex numbers, including their absolute value, graphing in the complex plane, and solving quadratic equations using the discriminant. Test your understanding of polynomial functions and inequalities as well.