What is the x-intercept of the line with the equation 4x + 2y = 12?
To find the x-intercept of the line given by the equation 4x + 2y = 12, we need to determine the point where the line crosses the x-axis. This occurs when the value of y is equal to zero. We start by substituting y = 0 into the equation: 4x + 2(0) = 12 This […]
What is the total cost of a pair of jeans that are priced $50.00 if there is 7.5% sales tax?
To determine the total cost of the jeans after applying the sales tax, you need to first calculate the amount of sales tax and then add it to the original price of the jeans. The formula to calculate sales tax is: Sales Tax = Price × Sales Tax Rate In this case, the price of […]
Given the exponential equation 4^x = 64, what is the logarithmic form of the equation in base 10?
To convert the exponential equation 4x = 64 into its logarithmic form in base 10, we need to follow a specific relationship between exponential and logarithmic expressions. The general principle is that if you have an equation in the form by = z, it can be expressed in logarithmic form as y = logb(z). In […]
What is the formula of diagonal of cube and cuboid?
The diagonal of a cube can be calculated using the formula: D = a√3 where D is the diagonal length and a is the length of one side of the cube. For a cuboid, the formula for the diagonal is slightly different: D = √(l² + w² + h²) In this case, D represents the […]
Write x² + 2x + 3 = 0 in the form x = a ± b
To rewrite the quadratic equation x² + 2x + 3 = 0 in the form x = a ± b, we will first need to use the quadratic formula: x = (-b ± √(b² – 4ac)) / 2a In our case, the equation is of the form ax² + bx + c = 0, where: […]
What is the root of the polynomial equation x^3 + 2x^2 + x – 18?
The roots of the polynomial equation x3 + 2×2 + x – 18 = 0 can be found using various methods such as factoring, synthetic division, or numerical methods. To find the roots, we can start by trying to factor the polynomial or use the Rational Root Theorem to find possible rational roots. Testing for […]
How many positive integers less than 1000 are divisible by both 7 and 11?
To find the number of positive integers less than 1000 that are divisible by both 7 and 11, we first need to determine their least common multiple (LCM). The LCM of 7 and 11 can be calculated since both are prime numbers. Therefore, the LCM is simply: LCM(7, 11) = 7 * 11 = 77 […]
Which explicit formula generates the infinite sequence 2, 9, 28, 65, 126?
The explicit formula for the given sequence can be derived by analyzing the pattern of the numbers. The sequence presented is 2, 9, 28, 65, 126, and can be noted as the series of values determined by the formula a(n) = n^3 – n + 2, where n is the term number starting from 1. […]
In how many ways can the letters in the word spoon be arranged?
The word ‘spoon’ consists of 5 letters where the letter ‘o’ appears twice. To find the number of distinct arrangements of the letters, we can use the formula for permutations of multiset: Number of arrangements = \( \frac{n!}{n_1! \cdot n_2! \cdot … \cdot n_k!} \) Here: n = total number of letters = 5 n1 […]
What is the differential equation for dy/dx = xe^y – 2x?
The differential equation given is: dy/dx = xe^y – 2x To understand this equation, we can analyze the right-hand side, which involves both an exponential function and a linear component. This equation signifies how the change in y (denoted as dy) with respect to x (denoted as dx) is influenced by the terms present on […]