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Questions and Answers
Which of the following sets is closed under subtraction?
Which of the following sets is closed under subtraction?
- The set of complex numbers ℂ (correct)
- The set of rational numbers ℚ (correct)
- The set of natural numbers ℕ
- The set of positive real numbers ℝ
What is the correct result of solving the equation $x^2 + 10x + 7 = 0$ by completing the square?
What is the correct result of solving the equation $x^2 + 10x + 7 = 0$ by completing the square?
- The solutions are $x = -5 \pm \sqrt{18}$ (correct)
- The solutions are $x = -5 \pm \sqrt{22}$
- The solutions are $x = -5 \pm \sqrt{34}$
- The solutions are $x = -5 \pm \sqrt{11}$
Which statement regarding numbers is true?
Which statement regarding numbers is true?
- The sum of a rational number and an irrational number is always irrational. (correct)
- The sum of a complex number and its conjugate is always equal to zero.
- The square of an irrational number is always rational.
- The product of any two complex numbers is always a real number.
What are the possible solutions for the equation $2x + 8x + 134 = 0$ using the completing the square method?
What are the possible solutions for the equation $2x + 8x + 134 = 0$ using the completing the square method?
Which of the following pairs of values represents the roots of the quadratic equation $x^2 + 4x + 22 = 67$?
Which of the following pairs of values represents the roots of the quadratic equation $x^2 + 4x + 22 = 67$?
What are the solutions to the equation $x^2 - 5x + 6 = 0$?
What are the solutions to the equation $x^2 - 5x + 6 = 0$?
Which of the following correctly simplifies the expression $-100 - 2 - 16$?
Which of the following correctly simplifies the expression $-100 - 2 - 16$?
What is the additive identity of $5 + 2i$?
What is the additive identity of $5 + 2i$?
What expression demonstrates the associative property of multiplication?
What expression demonstrates the associative property of multiplication?
What are the solutions to the equation $x^2 + 10x + 24 = 0$?
What are the solutions to the equation $x^2 + 10x + 24 = 0$?
Which complex number corresponds to $(-7 + 3i) + (-2 + 8i)$ in standard form?
Which complex number corresponds to $(-7 + 3i) + (-2 + 8i)$ in standard form?
Which sets are closed under multiplication?
Which sets are closed under multiplication?
What do the values equivalent to $-i$ include?
What do the values equivalent to $-i$ include?
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Study Notes
Solving Quadratic Equations
- Factoring is a method to solve for x when the equation is in the form ax^2 + bx + c = 0
- Factoring involves finding two numbers that multiply to c and add to b
- For example, 2x^2 - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0
- The solutions are x = 2 and x = 3
Simplifying Complex Numbers
- A complex number is made up of a real part and an imaginary part
- The imaginary unit, i, is defined as the square root of -1
- To simplify complex expressions, combine real and imaginary terms separately
- For example, (7+i) - (-2+8i) simplifies to 9 - 7i
Real Number Properties
- The associative property of multiplication states that the grouping of factors does not affect the product
- For example, ((1-i) * 8i) * -2i = (1-i) * (8i * -2i)
- The additive identity of a number is zero, meaning adding zero to any number results in the same number
- For example, 5 + 2i + 0 = 5 + 2i
Complex Number Properties
- The product of two complex numbers is not always a real number, for example (1+i)(1-i) = 2
- The square of an irrational number is not always rational, for example (√2)^2 = 2
- The sum of a rational number and an irrational number is always irrational
- The sum of a complex number and its conjugate is always equal to a real number, for example (2+3i)+(2-3i)=4
Solving Quadratic Equations by Completing the Square
- To solve by completing the square, manipulate the equation to have the form (x+a)^2 = b
- Move the constant term to the right side of the equation
- Take half of the coefficient of the x term, square it, and add it to both sides
- Take the square root of both sides and solve for x
- For example, x^2 + 10x + 7 = 0 can be rearranged to (x+5)^2 = 18 and solved for x
Sets
- The natural numbers (N) are positive integers 1, 2, 3, ...
- The rational numbers (Q) can be expressed as a fraction of two integers
- The real numbers (R) include all rational and irrational numbers
- The complex numbers (C) include all real and imaginary numbers
Closure Property
- A set is closed under an operation if the operation on any two elements of the set results in an element within the same set
- For example, the set of real numbers is closed under addition because adding two real numbers always results in another real number
- The set of natural numbers is not closed under subtraction because subtracting two natural numbers may result in a negative number, which is not a natural number
- The set of complex numbers are closed under subtraction and multiplication.
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