To write the equation of a circle, we use the standard form of the equation, which is given as: (x – h)2 + (y – k)2 = r2 In this equation: (h, k) represents the coordinates of the center of the circle. r is the radius of the circle. Given that the center of the … Read more
To calculate the length of the arc (L) in a circle, you can use the formula: L = (θ / 360) * 2πr Where: L = length of the arc θ = angle in degrees r = radius of the circle π (pi) = approximately 3.14159 In this case: θ = 30 degrees r = … Read more
To solve the equation x² + 5x = 0 using the quadratic formula, we start by recognizing that this is a quadratic equation in the standard form ax² + bx + c = 0. In this case, we can identify the coefficients as follows: a = 1 (the coefficient of x²) b = 5 (the … Read more
To solve for the exact value of x using special right triangles, we can leverage the properties of the two primary types of special right triangles: the 45-45-90 triangle and the 30-60-90 triangle. 1. 45-45-90 Triangle A 45-45-90 triangle has angles of 45°, 45°, and 90°. The sides opposite the 45° angles are equal, and … Read more
Synthetic division is a simplified method of polynomial long division, allowing us to divide a polynomial by a linear factor in a more straightforward way. Let’s break down the process step by step for the polynomial 2x³ + 0x² + 3x + 22, assuming we want to divide it by x – r, where r … Read more
Finding the General Solution To solve the differential equation \( x \frac{dy}{dx} + y = x^2 \sin x \), we’ll follow a systematic approach to separate the variables and simplify.’ Step 1: Rearranging the Equation We start by rearranging the given differential equation to isolate \( \frac{dy}{dx} \). First, we can move \( y \) … Read more
When you draw all the diagonals from a single vertex of a hexagon, you can create several triangles. Let’s break it down step by step. A hexagon has six sides, and when you select any vertex to start drawing diagonals, you will connect that vertex to non-adjacent vertices. Each vertex connects to the other vertices … Read more
To determine the expression that is equivalent to the product of m and n, let’s first substitute the given expressions: We have: m = x2 + 3 n = 5x + 9 Now, we need to multiply m and n: mn = (x2 + 3)(5x + 9) To perform the multiplication, we will use the … Read more
To find the equation in point-slope form of the line that passes through the points (2, 5) and (2, 3), we first need to determine the slope (m) of the line. The formula for calculating the slope between two points (x_1, y_1) and (x_2, y_2) is: m = (y_2 – y_1) / (x_2 – x_1) … Read more
To find the derivatives dy/dx and d²y/dx² for the equation x² sin(t) + y³ cos(t) = 0, we will need to use implicit differentiation and the chain rule. Let’s break this down step by step: Step 1: Implicit Differentiation We start with the equation: x² sin(t) + y³ cos(t) = 0 Taking the derivative with … Read more