How to Find the Roots of the Quadratic Equation 1x² + 1x + 2 = 3?
To find the roots of the quadratic equation, we start by rewriting it in standard form. The equation given is: 1x² + 1x + 2 = 3 First, we subtract 3 from both sides to set the equation to zero: 1x² + 1x + 2 – 3 = 0 This simplifies to: x² + x […]
What is the slope of the line passing through (3, 5) and (5, 3)?
The slope of a line is calculated using the formula: Slope (m) = (y₂ – y₁) / (x₂ – x₁) In this case, the two points given are (3, 5) and (5, 3). Here, (x₁, y₁) = (3, 5) and (x₂, y₂) = (5, 3). Now, substituting the values into the formula: m = (3 […]
What are the first ten terms of a sequence whose first term is 10 and whose common difference is 2?
The first ten terms of the sequence can be found by starting with the first term and repeatedly adding the common difference. In this case, the first term is 10, and the common difference is 2. We can calculate the terms as follows: 1st term: 10 2nd term: 10 + 2 = 12 3rd term: […]
Plot the point whose polar coordinates are given then find the cartesian coordinates of the point 3, 3π/2
To plot the point given in polar coordinates, we start with the values provided: the radius is 3 and the angle is 3π/2. In polar coordinates, the general form is (r, θ), where r represents the distance from the origin and θ represents the angle measured from the positive x-axis. The angle 3π/2 radians corresponds […]
How to Solve the Initial Value Problem: xy² dx + 2xy x² + 5 dy = 0, with y(1) = 1?
To solve the initial value problem given by the differential equation: xy² dx + 2xy x² + 5 dy = 0 with the initial condition y(1) = 1, we start by rearranging the equation: 1. First, rewrite the equation in a more standard form: dy/dx = – (xy²)/(2xy x² + 5) 2. To separate variables, […]
What function would you use to calculate the total interest paid for the first year of a mortgage?
To calculate the total interest paid for the first year of a mortgage, you would typically use the PMT function in conjunction with the IPMT function in Excel or similar spreadsheet software. The PMT function helps you determine the monthly payment amount based on the loan principal, interest rate, and total number of payments. However, […]
During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. How many of each would you need to buy?
To determine how many packs of color pencils and crayons you would need to buy, we first need to consider how many items you intend to have. The number of packs you require will depend on the total quantity of each item you wish to own. If we take a specific example: let’s say you […]
How can one sixth x 5 one fifth x 2 be set up as a system of equations?
To set up the expression “one sixth x 5 one fifth x 2” as a system of equations, we first need to break down the components of the expression into manageable parts. The phrase seems to combine fractional coefficients with a variable, which we will denote as x. 1. The expression begins with one sixth, […]
What is the Golden Ratio?
The Golden Ratio, often represented by the Greek letter phi (φ), is a mathematical concept that has fascinated artists, architects, and mathematicians for centuries. It is approximately equal to 1.6180339887. To understand the Golden Ratio, consider a line segment divided into two parts, A and B, such that the ratio of the entire length (A […]
How can I use the Rational Zeros Theorem to find all possible rational zeros of the function f(x) = 2x³ + 8x² – 7x – 8?
The Rational Zeros Theorem states that any rational solution, or zero, of a polynomial function can be expressed as a fraction p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. For the function f(x) = 2x³ + 8x² – 7x – 8, we first identify […]