🧮 algebra

1. **Stating the problem:** Three girls initially paid a total of $300 for a motel room ($100 each). Later, they were supposed to be charged only $250, so the clerk gives $50 to th

1. The problem is to find the x- and y-intercepts of a function and identify extrema (maxima or minima). 2. The function described is not specified, so let's consider a generic exa

1. The problem is to solve the equation $$3x^2 - 12x + 9 = 0$$. 2. First, we can simplify the equation by dividing all terms by 3:

1. The problem involves a number machine with two steps: first an operation \(An!\), then multiplication by 5, giving the output. 2. Given the input is 15 and the output is 30, we

1. The problem gives the number of dots for the first three terms: term 1 has 5 dots, term 2 has 8 dots, and term 3 has 11 dots. 2. We observe that the number of dots increases by

1. We are given a sequence where the 1st pattern has 5 dots, the 2nd has 8 dots, and the 3rd has 11 dots. We want to find an expression for the number of dots in the $n$th pattern.

1. The problem asks for a rule for the number of sticks in the nth pattern, given: 1st pattern has 5 sticks, 2nd pattern has 9 sticks, 3rd pattern has 13 sticks. 2. Observe the pat

1. **State the problem:** Find the geometric mean between the pairs of numbers given: I. 9 and 16

1. The problem presents a pattern of dots where the 1st pattern has 7 dots, the 2nd has 10 dots, and the 3rd has 13 dots. 2. Observe the differences between consecutive terms: $10

1. **Problem (a): Solve the indicial equation** $2^x + 2^{x-1} = 48$ 2. Rewrite $2^{x-1}$ as $2^x \cdot \frac{1}{2}$ since subtracting 1 from the exponent divides by 2.

1. **Simplify each algebraic expression:** i) Simplify $7a + 3b - 5a + b - 6$

1. **State the problem:** We need to find the first five terms and the tenth term of a geometric sequence where the first term $a=3$ and the common ratio $r=\frac{-1}{2}$. 2. **Rec

1. Solve for $x$ in each indicial equation. **a.** $2^x + 2^{x-1} = 48$

1. **Problem a:** Solve the indicial equation $$2^x + 2^{x-1} = 48$$ Step 1: Rewrite the terms to factorize:

1. Problem: Factorize the polynomial $x^3 - x^2 - 17x - 15$ given it can be expressed as $(x + 3)(x^2 + bx + c)$. Find $b$ and $c$. Step 1: Expand the right-hand side:

1. **State the problem:** Lisa is 7 years old. Her aunt says, "When you reach my current age, I will be 41 years old." We need to find Lisa's aunt's current age. 2. **Define variab

1. **State the problem:** There are 25 motorcycles, scooters, and cars in total, and together they have 68 wheels. We need to find how many cars there are. 2. **Define variables:**

1. **State the problem:** Solve for $\Delta x$ given the equation $$\frac{\sqrt{45}}{\Delta x} = \frac{6x}{2\sqrt{5}}.$$\n\n2. **Simplify square roots and fractions:** \n$\sqrt{45}

1. **Stating the problem:** Solve the equation $$\frac{3y - 2}{8} - \frac{4}{3} = \frac{y}{2} - \frac{11}{24}.$$\n\n2. **Rewrite the equation for clarity:** $$\frac{3y - 2}{8} - \f

1. Let's start by understanding the request: you asked for examples, but did not specify the topic or type of math problem you want examples of. 2. Since the request is vague, I wi

1. The problem asks for the standard equation of a circle with radius $\frac{5}{3}$ units and center at $(9.2, -7.4)$. 2. Recall the standard equation of a circle with center $(h,k