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To find the value of z such that both 9 and -9 are solutions to the equation x² + z = 103, we start by substituting the values of x into the equation. 1. First, let’s substitute x = 9:     x² + z = 103     9² + z = 103     81 + z = 103 … Read more

To solve the equation 3x² + 18x + 21 = 0 by completing the square, the first step involves simplifying the equation. Here’s how to start: Factor out the coefficient of x²: Since the coefficient of x² is 3, we want to factor it out from the first two terms: 3(x² + 6x) + 21 … Read more

To solve this problem, we note that when one variable varies inversely with another, it can be expressed with the equation: y = k/x Where: y is the dependent variable x is the independent variable k is the constant of variation From the problem statement, we know two things: y = 40 x = 5 … Read more

To compute the limit \( \lim_{h \to 0} \frac{f(5 + h) – f(5)}{h} \), given the function \( f(x) = 12x \), we first need to substitute the values into the limit expression. 1. **Evaluate \( f(5) \):** Let’s calculate \( f(5) \): \( f(5) = 12 \times 5 = 60 \) 2. **Evaluate \( … Read more

To determine the factors of the expression 2xy + 8y + 8x + 32, we first need to rearrange and group the terms in a way that allows for factoring. Step 1: Group like terms. We can group the first two terms and the last two terms: (2xy + 8y) + (8x + 32) Step … Read more

To convert the given equation of the sphere into standard form and find the center and radius, we will complete the square for each of the variables (x, y, and z) in the equation: The equation we have is: x² + y² + z² + 2x – 6y + 8z – 1 = 0 First, … Read more

To find the value of f(13) where f(x) = log₂(x), we need to substitute x with 13. So, we are calculating: f(13) = log₂(13) The logarithm base 2 of a number tells us the power to which 2 must be raised to obtain that number. In this case, we want to find out what exponent … Read more

To solve for the measures of angles A, B, and C in triangle ABC, we can use the information provided and some basic principles of geometry. 1. **Defining Variables:** Let’s denote the measure of angle A as x degrees. According to the problem, we know: Angle B = 3 × angle A = 3x degrees … Read more

To find the sixth term in the binomial expansion of the expression (5y + 3)10, we can use the binomial theorem. The binomial theorem states that: (a + b)n = = (nCr) * an-r * br, where nCr is the binomial coefficient, n is the power, a is the first term, b is the second … Read more

The end behavior of a polynomial function is determined primarily by its leading term, which is the term with the highest degree. In the case of the polynomial function f(x) = 2×3 + 26x + 24, the leading term is 2×3. Since the degree of the leading term is 3 (an odd number) and the … Read more