How do you simplify the rational expression \( \frac{k^2}{2k^2 + 4k + 5} \) and what are the restrictions on the variable?
To simplify the rational expression \( \frac{k^2}{2k^2 + 4k + 5} \), we first look for factors in the denominator. 1. **Factor the denominator**: The quadratic expression in the denominator is \( 2k^2 + 4k + 5 \). We can check if it can be factored: The discriminant of the quadratic \( b^2 – 4ac […]
What is the equation of a line when the slope is undefined and the x-intercept is 7?
When the slope of a line is undefined, it indicates that the line is vertical. A vertical line has the same x-coordinate for every point on the line, which means it doesn’t rise or fall as you move along the y-axis. Given that the x-intercept is 7, this tells us that the line crosses the […]
The sequence an is defined by a0 = 1 and an = 2an-1 + 2 for n ≥ 0. What is the value of a3?
To find the value of a3, we first need to calculate the earlier terms of the sequence based on the definition provided. 1. Given that a0 = 1. 2. Now, let’s find a1: a1 = 2a0 + 2 = 2(1) + 2 = 2 + 2 = 4. 3. Next, we calculate a2: a2 = […]
What graph correctly solves the system of equations below: y = x² + 1 and y = 2x² – 4?
To solve the system of equations given by y = x² + 1 and y = 2x² – 4, we first need to find the points where the two graphs intersect. This can be done by setting the equations equal to each other: x² + 1 = 2x² – 4 Rearranging this equation leads to: […]
Is Multiplication Associative for Rational Numbers?
Yes, multiplication is associative for rational numbers. This means that when you multiply three or more rational numbers, the way in which the numbers are grouped does not affect the product. To explain this, let’s take three rational numbers: a, b, and c. According to the associative property of multiplication, we can say: (a × […]
How to Find ab if a = 3, b = 14, and the Angle Between a and b is 45 degrees?
To find the product of a and b, denoted as ab, with the values given, we can use the properties of vectors. Here, we have: a = 3 b = 14 Angle between a and b = 45 degrees The formula to find the product of two vectors considering the angle between them is: ab […]
Which of the following is a zero for the function f(x) = x^3 – 3x^2 – 7x + 5?
To find a zero for the function f(x) = x3 – 3×2 – 7x + 5, we need to solve the equation f(x) = 0. This means we are looking for values of x that make the equation equal to zero. We can test potential zeros by substituting different values of x into the function. […]
Which of the following could be the lengths of the sides of a 45-45-90 triangle?
A 45-45-90 triangle is an isosceles right triangle, and its sides have a specific ratio. In such triangles, the lengths of the legs are equal, and the hypotenuse is √2 times the length of a leg. For example, if the lengths of the legs are both ‘x’, the length of the hypotenuse would be ‘x√2’. […]
What is the arc length of an angle of π/3 radians formed on the unit circle?
The arc length of an angle on a circle can be found using the formula: Arc Length = r × θ Where r is the radius of the circle and θ is the angle in radians. For a unit circle, the radius r is 1. In this case, the angle given is π/3 radians. Substituting […]
What is the equation of the line perpendicular to the line y = 49x + 2 that passes through the point (4, 3)?
To find the equation of the line that is perpendicular to the line y = 49x + 2 and passes through the point (4, 3), we first need to determine the slope of the given line. The equation of the line is in the slope-intercept form y = mx + b, where m represents the […]