Which of the following is a conjugate for 10 + 3i?
The conjugate of a complex number is formed by changing the sign of the imaginary part. For the complex number 10 + 3i, its conjugate is 10 – 3i. In more detail, if you have a complex number represented as a + bi, where a is the real part and b is the imaginary part, […]
What are the values of the angles in a triangle defined by 2x, 3x, and 4x degrees?
To find the value of x in the triangle with angles 2x, 3x, and 4x degrees, we start by recalling that the sum of the angles in any triangle is always 180 degrees. We can set up the equation as follows: 2x + 3x + 4x = 180 This simplifies to: 9x = 180 Now, […]
What is the solution set of the system using substitution 3x + 2y = 7 and y = 3x + 11?
To solve the system of equations using substitution, we start with the equations: 3x + 2y = 7 y = 3x + 11 We can substitute the expression for y from the second equation into the first equation. This gives us: 3x + 2(3x + 11) = 7 Next, we distribute the 2: 3x + […]
Find the Exact Length of the Curve y = 8x^3 + 2, for 0 ≤ x ≤ 1
To find the exact length of the curve described by the function y = 8x³ + 2 over the interval from x = 0 to x = 1, we will use the formula for the arc length of a curve represented as y = f(x). The formula for the arc length L from x = […]
What is the measure of arc BC in circle O when line segments AC and BD are diameters?
To solve for the measure of arc BC in circle O, we start by analyzing the information given. We know that line segments AC and BD are diameters of the circle. Therefore, arcs AB and DC are related to these diameters. From the problem statement, we have: Measure of arc AB = 3x – 70 […]
Using the following equation find the center and radius x² + 2x + y² – 4y = 20
To find the center and radius of the given equation, we first need to rewrite it in the standard form of a circle’s equation, which is: (x – h)² + (y – k)² = r², where (h, k) is the center and r is the radius. The given equation is: x² + 2x + y² […]
How to Solve the System Algebraically: 7x + 2y = 4 and 5y = 3x + 10?
To solve the system of equations algebraically, we have the following two equations: Equation 1: 7x + 2y = 4 Equation 2: 5y = 3x + 10 First, let’s rearrange Equation 2 to express y in terms of x. We can do this by isolating y: 5y = 3x + 10 Dividing both sides by […]
How to Find the Point on the Line y = 5x + 2 Closest to the Origin?
To find the point on the line y = 5x + 2 that is closest to the origin (0, 0), we can use the concept of distance from a point to a line. The distance D from a point (x, y) to the origin is given by: D = √(x² + y²) However, since we […]
Which of the following exponential functions goes through the points (1, 20) and (2, 80)?
To determine which exponential function passes through the points (1, 20) and (2, 80), we can represent an exponential function in the form of y = abx, where a is the initial value and b is the growth factor. From the first point (1, 20), we substitute into the equation: 20 = ab1 → 20 […]
Find the Quotient of 2x^4 + 3x^3 + 6x^2 + 11x + 8 ÷ (x + 2)
To find the quotient when dividing the polynomial 2×4 + 3×3 + 6×2 + 11x + 8 by (x + 2), we can use polynomial long division. 1. **Set up the division:** Write 2×4 + 3×3 + 6×2 + 11x + 8 under the long division symbol and (x + 2) outside. 2. **Divide the […]