How do you simplify the expression 5 + 3i + 2i?
To simplify the expression 5 + 3i + 2i, we start by combining the like terms that contain the imaginary unit i. First, identify the coefficients of i in the expression: 3i 2i Next, add the coefficients together: 3 + 2 = 5 Now, we can rewrite the expression by combining the real part and […]
Each leg of a 45-45-90 triangle measures 12 cm. What is the length of the hypotenuse?
In a 45-45-90 triangle, the legs are of equal length, and the hypotenuse is always equal to the length of a leg multiplied by the square root of 2. Given that each leg of the triangle measures 12 cm, we can calculate the hypotenuse using the formula: Hypotenuse = Leg × √2 Substituting the values […]
Let f(x) = 8x³ + 28x + 61 and g(x) = 2x + 5. Find f(g(x)).
To find f(g(x)), we first need to understand what each function represents. The function f(x) is given as: f(x) = 8x³ + 28x + 61 And the function g(x) is: g(x) = 2x + 5 Now, to find f(g(x)), we will substitute g(x) into the function f. This means wherever we see x in f(x), […]
What are the present ages of A and B if five years ago A was four times the age of B, and five years hence A will be twice the age of B?
Let the present age of A be A years and the present age of B be B years. According to the problem: Five years ago, A’s age was A – 5 and B’s age was B – 5. At that time, the relationship was: A – 5 = 4(B – 5) Expanding the equation: A […]
How to Write an Equation for the Translation of y = 4x with Asymptotes x = 7 and y = 6?
To find the equation of the translated function, we can start with the basic function y = 4x. The goal is to translate this function so that it has vertical and horizontal asymptotes at x = 7 and y = 6, respectively. The given asymptotes indicate that the graph of the function must be shifted […]
Using the given points determine the slope of the line (3, 0) and (2, 7)
To determine the slope of the line passing through the points (3, 0) and (2, 7), we use the formula for slope, which is: m = (y2 – y1) / (x2 – x1) Here, (x1, y1) = (3, 0) and (x2, y2) = (2, 7). Substituting the values into the formula: m = (7 – […]
Find the Mean, Median, and Mode of the Data Set
To find the mean, median, and mode of a data set, we need to follow a few simple steps for each measure of central tendency. Mean The mean is calculated by adding all the numbers in the data set and then dividing by the total number of values. For example, if our data set is […]
What is the new volume of a gas when its temperature changes from 27°C to 30°C?
To find the new volume of the gas when the temperature changes, we can use Charles’s Law, which states that the volume of a gas is directly proportional to its temperature when pressure is constant. The formula for Charles’s Law is: V1 / T1 = V2 / T2 Where: V1 is the initial volume (605 […]
Identify the Type of Conic Section from the Equation: x² + 4y – 2y². Find the Vertices and Foci.
The given equation is x² + 4y – 2y². To identify the type of conic section, we can rearrange the equation into the standard form. First, let’s rewrite it: 2y² – 4y – x² = 0. This can be expressed in the standard form of a conic section. To do this, we can complete the […]
Which graph shows the axis of symmetry for the function f(x) = x^2 + 2x + 1?
The function f(x) = x² + 2x + 1 is a quadratic function that can be rewritten in vertex form to find its axis of symmetry. To do this, we can complete the square or use the standard formula for the axis of symmetry of a quadratic function. For the general form of a quadratic […]