What is the 8th term of the geometric sequence where a1 is 256 and a3 is 16?
To find the 8th term of the geometric sequence, we first need to establish the common ratio of the sequence. We know: The first term, a1, is 256. The third term, a3, is 16. In a geometric sequence, each term can be expressed as: an = a1 × r(n-1) Where r is the common ratio. […]
In circle D, angle ADC measures 7x + 2, and arc AC measures 8x + 8. What is the measure of angle ABC?
To find the measure of angle ABC, we first need to understand how the angle relates to the arc in a circle. The measure of angle ABC is half the measure of the intercepted arc AC. We know the measure of arc AC is given as: Arc AC = 8x + 8 According to the […]
If f(x) = 3x + 4 and g(x) = 2, solve for the value of x for which f(x) = g(x)
To find the value of x for which f(x) equals g(x), we start by setting the two functions equal to each other: f(x) = g(x) 3x + 4 = 2 Next, we will isolate x. First, we can subtract 4 from both sides: 3x = 2 – 4 3x = -2 Now, divide both sides […]
What is the equation of a vertical line passing through the point (5, 1)?
A vertical line has the same x-coordinate for all points on the line. Since the line passes through the point (5, 1), the x-coordinate is 5. Therefore, the equation of the vertical line is simply: x = 5 This means that for every point on this line, the x-value will always be 5 while the […]
How do you solve the equation 2x² + 12x + 14 = 0?
To solve the quadratic equation 2x² + 12x + 14 = 0, we can use the quadratic formula, which is: x = (-b ± √(b² – 4ac)) / 2a In our case, the coefficients are: a = 2 b = 12 c = 14 First, we need to calculate the discriminant (b² – 4ac): b² […]
How do I solve the equations 2x + 13y = 4 and x + 8y = 9?
To solve the equations 2x + 13y = 4 and x + 8y = 9, we can use the substitution or elimination method. Here, we’ll use the elimination method for clarity. First, let’s rewrite the second equation to isolate x
If f(x) = x^2 and g(x) = x^4, find f(g(x))
To find f(g(x)), we need to substitute g(x) into the function f(x). We know that: f(x) = x^2 g(x) = x^4 Now, we can substitute g(x) into f(x): f(g(x)) = f(x^4) This means we replace x in f(x) with x^4: f(g(x)) = (x^4)^2 When we simplify this, we get: (x^4)^2 = x^{4*2} = x^8 So, […]
What are all possible two-digit numbers that can be formed using 3, 7, and 9 with repetition allowed?
To find all possible two-digit numbers that can be formed using the digits 3, 7, and 9 with repetition allowed, we follow a simple combinatorial approach. For two-digit numbers, the first digit (tens place) can be any of the three digits: 3, 7, or 9. The second digit (units place) can also be any of […]
What algebraic property is demonstrated in the example: 2xy + x^2y?
The algebraic property demonstrated in the expression 2xy + x^2y is known as the Distributive Property. To break it down, we can factor out the common term y from both parts of the expression: 2xy + x^2y = y(2x + x^2) This shows how the Distributive Property allows us to combine terms by factoring out […]
Write an equation for the translation of the function y = cos(x) translated 6 units up
To translate the function y = cos(x) 6 units upward, you need to add 6 to the entire function. This vertical shift increases the output of the function by 6 units for every value of x. Thus, the new equation after the translation becomes: y = cos(x) + 6 This means that for every point […]