Complex Numbers and Equations

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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?

Combine like terms, then if it's multiplying, work left to right. Signup and view all the answers

What is the value of i^2?

-1 Signup and view all the answers

How do you find the discriminant value of a quadratic equation and determine the type of solutions?

Use the formula b^2 - 4ac. If the answer is a negative root, it's complex; if the answer is a root of 0, there's only 1 solution; if the answer is a whole number root, it's real. Signup and view all the answers

How do you write a polynomial function of least degree with integral coefficients given the zeros?

Use the complex conjugates, then multiply with each set. Signup and view all the answers

What is the process to sketch the graph of an inequality?

Plug it into the calculator. If it's not equal to, it's a dotted line. It has a parabola. Signup and view all the answers

How do you graph each inequality?

Plug it into the calculator. It's the shape of a V. If it's not equal to, it's a dotted line. Signup and view all the answers

How do you solve each inequality and graph its solution on the number line when it's greater than?

Set them into two equations, but flip the sign for the negative answer. Solve algebraically. Signup and view all the answers

How do you solve each inequality and graph its solution on the number line when it's less than?

The answer should be negative and positive on both sides. Get m by itself. Don't flip the signs unless you divide by a negative. Signup and view all the answers

What is the method to solve each quadratic inequality and graph the solutions on the number line?

Set the problem equal to 0. Factor. Use ZPP to find the 2 solutions. Signup and view all the answers

How do you solve each equation by factoring?

Plug in the numbers to see if the equation is true. Signup and view all the answers

What is the process to solve each equation by taking square roots?

Get the x value on one side. Square root both sides. Signup and view all the answers

How do you solve each equation using the quadratic formula?

Do x = -b ± √(b² - 4ac)/2a Signup and view all the answers

What steps do you take to sketch the graph of each function and identify the roots/zeros if possible?

Plug it into a calculator. If the parabola does not touch the x-axis, it has no solutions; touches once, one solution; touches twice, two solutions. Name them. Signup and view all the answers

What is the process to simplify square roots?

Take out perfect squares and leave that answer on the outside while still having the other square root in its place. Signup and view all the answers

What should you do if the equation has square roots on both ends?

Square it on both sides to cancel it out. Solve. Signup and view all the answers

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.