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Questions and Answers
What is the least common multiple (LCM) of the numbers 75, 80, and 90?
What is the absolute value of the complex number 3 + 4i?
What are the solutions for the equation $ ext{sqrt}(20 - x) = x$?
What is the value of x that satisfies the equations x + y = 7 and x² + 2y² = 34?
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Determine the product of the roots of the equation $x^2 - 4x + 3m = 0$ given that one root exceeds the other by 2.
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Which of the following represents a correctly transformed equation from $x + y = 7$ into the form $x^2 + 2y^2 = 34$?
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What is the correct method to solve for the roots of the equation $x^2 + x - 20 = 0$?
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How can one find the ordered triples of real numbers (x, y, z) that satisfy the equations given?
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How many positive real roots does the polynomial equation $x^7 + 14x^3 + 16x + 30x - 560 = 0$ have?
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What is the discriminant of the quadratic equation $4x^2 - 8x + 5 = 0$?
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What is the natural logarithm of 'e'?
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What is the simplified form of $(0.001)^{ rac{2}{3}}$?
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Which factorization is correct for the expression $3x^3 - 3x^2 - 18x$?
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Which of the following correctly describes the number 8 + 0i?
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For the equation $4x^3 - 2x^2 + x - 3 = 0$, how many positive real roots are there?
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Which of the following options represents the proper simplification of $ rac{3x - 1}{x + 3} imes rac{x^2 - 1}{x^2 + 3x + 2} imes rac{x + 2}{2x - 3}$?
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What is one of the values of x that satisfies the equation $x^{-3} - 9x^{-3/2} + 8 = 0$?
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What is the surface area of a hollow cone if r = 3.0 cm and L = 8.5 cm?
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What values of x and y satisfy the equations $8^{(x)} = 2^{(y + 2)}$ and $16^{(3x - y)} = 4^y$?
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What are the roots of the cubic equation $x^3 - 8x - 3 = 0$?
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What is the solution to the equation $50x^2 + 5(x - 2)^2 = -1$?
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Which of the following numbers results in 1 when subjected to the happy number process?
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When solving $1.4 = ( rac{0.0613}{x})^{-3.2}$, what is the value of x?
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What is the approximate value of x that satisfies the equation $17.3 = e^{1.1x}$?
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Study Notes
Finding the Least Common Multiple (LCM)
- The LCM of 75, 80, and 90 is 3600 cm.
- This is calculated by finding the prime factorization of each number and then taking the highest power of each prime factor.
Absolute Value of Complex Numbers
- The absolute value of a complex number (a + bi) is calculated as the square root of (a² + b²).
- The absolute value of 3 + 4i is 5.
Solving Quadratic Equations
- The quadratic equation x² + x - 20 = 0 can be factored into (x + 5)(x - 4) = 0.
- This leads to solutions of x = -5 and x = 4.
Solving Systems of Equations
- The system of equations x + y = 7 and x² + 2y² = 34 can be solved by substituting y from the first equation into the second equation.
- The solution is x = 4.
Solving Equations with Multiple Variables
- The system of equations (x + y)(x + y + z) = 384, (y + z)(x + y + z) = 288, and (x + z)(x + y + z) = 480 can be solved by adding all three equations together.
- This leads to x + y + z = ±24.
- By substituting this value back into each of the original equations, we find the solution sets are (x = ±12, y = ±4, z = ±8).
Solving Quadratic Equations with a Given Relationship Between Roots
- If one root of the quadratic equation x² - 4x + 3m = 0 exceeds the other by 2, we can use the relationship between the roots to find the value of m.
- The sum of the roots is 4, and since one root is 2 more than the other, we can find the roots as 1 and 3.
- The product of the roots is 3, so m = 1.
Descartes' Rule of Signs
- The number of positive real roots of a polynomial equation is equal to the number of sign changes in the polynomial or less than that number by an even positive integer.
- The polynomial $x^7 + 14x^3 + 16x + 30x - 560 = 0$ has one sign change, indicating that it has one positive real root.
Discriminant of a Quadratic Equation
- The discriminant of the quadratic equation 4x² - 8x + 5 = 0 is -16.
- This is calculated using the formula D = b² - 4ac, where a = 4, b = -8, and c = 5.
Solving Cubic Equations
- The cubic equation $4x^3 - 2x^2 + x - 3 = 0$ has one positive real root.
- This can be found using a calculator or by using numerical methods.
Natural Logarithm of e
- The natural logarithm of e (ln e) is equal to xy, where y is the natural logarithm of e.
Exponent Operations
- The expression $(0.001)^\frac{2}{3}$ can be simplified as antilog [log 0.001]$\frac{2}{3}$.
Factoring Expressions
- The expression $3x^3 - 3x^2 - 18x$ can be factored as 3x(x - 3)(x + 2).
Classifying Numbers
- 8 + 0i is a real number.
Simplifying Fractions
- The fraction $\frac{3x - 1}{x + 3} \times \frac{x^2 - 1}{x^2 + 3x + 2} \times \frac{x + 2}{2x - 3}$ can be simplified to $\frac{1}{2x - 3}$.
Solving Equations with Fractional Exponents
- The equation $x^{-3} - 9x^{-3/2} + 8 = 0$ can be solved by substituting x = 8, 1.
- The solution is x = 1/4.
Solving Equations with Exponential Terms
- The equation 1.4 = $(\frac{0.0613}{x})^{-3.2}$ can be solved using a calculator.
- The solution is x = 0.04751.
Surface Area of a Cone
- The surface area of a hollow cone is given by the formula A = πrL, where r is the radius and L is the slant height.
- When r = 3.0 cm and L = 8.5 cm, the surface area is 80.1 cm².
Solving Exponential Equations with Multiple Variables
- The system of equations $8^{(x)} = 2^{(y + 2)}$ and $16^{(3x - y)} = 4^y$ can be solved by equating the exponents after expressing all terms in the same base.
- The solutions are x = 2 and y = 4.
Solving Cubic Equations
- The cubic equation $x^3 - 8x - 3 = 0$ has roots x = 3, -0.382, -2.62.
Solving Exponential Equations with a Constant Term
- The equation $17.3 = e^{1.1x}$ can be solved using a calculator.
- The solution is x = 2.6.
Determining the Solution of a Quadratic Equation
- The equation $50x^2 + 5(x - 2)^2 = -1$ has no real solutions.
- This is because squaring any real number will always result in a non-negative value.
Happy Numbers
- A happy number is a number that eventually reaches 1 when repeatedly replacing it with the sum of the squares of its digits.
- 100 is a happy number because $1^2 + 0^2 + 0^2 = 1$.
Determining the Number of Real Roots in a Polynomial Equation
- The equation $x^7 + 14x^3 + 16x + 30x - 560 = 0$ has one real solution.
- This can be determined using Descartes' Rule of Signs, which states that the number of positive real roots is equal to the number of sign changes in the polynomial or less than that number by an even positive integer.
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Description
This quiz covers essential algebra concepts including finding the least common multiple (LCM), calculating the absolute value of complex numbers, and solving quadratic equations. Additionally, it explores systems of equations and equations with multiple variables. Test your understanding of these fundamental topics!
