What is the radius of a circle whose equation is x² + y² – 8x – 6y – 21 = 0 units?
To find the radius of the circle given by the equation x² + y² – 8x – 6y – 21 = 0, we first need to rewrite the equation in standard form. The standard form of a circle’s equation is: (x – h)² + (y – k)² = r² where (h, k) is the center […]
What is a factor of f(x) = 5x³ + 24x² + 75x + 14?
To find a factor of the polynomial f(x) = 5x³ + 24x² + 75x + 14, we can use the Rational Root Theorem, synthetic division, or polynomial long division. We can start by testing some possible rational roots based on the constant term (14) and the leading coefficient (5). The candidates for potential rational roots […]
Find the product of 4x, 32x², 7x, and 1
To find the product of the numbers and variables, we multiply them together. First, let’s write down the expression we need to calculate: 4x * 32x² * 7x * 1 We can multiply the coefficients (the numbers in front) together and also combine the variables: 1. Multiply the coefficients: 4 * 32 * 7 * […]
Given the parent functions f(x) = log2(3x + 9) and g(x) = log2(x + 3), what is f(g(x))?
To find f(g(x)), we need to substitute g(x) into the function f(x). Let’s start by writing down what each function looks like. We have: f(x) = log2(3x + 9) g(x) = log2(x + 3) Now, we can substitute g(x) into f(x): f(g(x)) = f(log2(x + 3)) This implies we need to replace the x in […]
How to Determine if Two Lines are Parallel Given Their Equations
To determine if two lines are parallel, you need to compare their slopes. The slope of a line can be found from its equation, which is typically in the form of either y = mx + b or Ax + By + C = 0. 1. **Find the Slope from the Equation**: If the line […]
Find three positive numbers whose sum is 100 and whose product is a maximum
To find three positive numbers whose sum is 100 and whose product is maximized, we can use the method of Lagrange multipliers or simply apply some algebraic insights. Let’s denote the three numbers as x, y, and z. According to the problem, we have: x + y + z = 100 To maximize the product […]
If an integer is randomly selected from all positive 2 digit integers, what is the probability that the integer chosen has a specific property?
To determine the probability of randomly selecting a certain type of integer from all positive two-digit integers, we first need to establish our sample space and the conditions that define the integers we’re interested in. The range of positive two-digit integers is from 10 to 99, inclusive. This gives us a total of: 99 – […]
Given the arithmetic sequence an = 4n + 1, what is the domain for n?
The domain for n in the arithmetic sequence an = 4n + 1 is all real numbers. This means that you can choose any real number for n, and the sequence will still produce a valid output. In an arithmetic sequence, n typically represents the term number, which can take on integer values starting from […]
What are the real and complex solutions of the polynomial equation x³ – 64 = 0?
To solve the polynomial equation x³ – 64 = 0, we first recognize that this can be rewritten as x³ = 64. The next step is to find the cube roots of 64. Since 64 = 4³, we know that one of the solutions is x = 4. However, a cubic equation can have up […]
What is an equation for the line with slope 23 and y intercept 9?
To find the equation of a line, we can use the slope-intercept form, which is given by the formula: y = mx + b In this formula: y represents the dependent variable (the value of y on the y-axis), m represents the slope of the line, x represents the independent variable (the value of x […]