🧮 algebra

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

1. **Solve the equation** $\frac{4p - 2}{5} = \frac{p + 1}{5} - 3$. 2. Multiply both sides by 5 to clear the denominators:

1. The problem is to graph the circle given by the equation $$(x + 2)^2 + (y - 6)^2 = 9.$$\n\n2. This equation is in the standard form of a circle's equation: $$(x - h)^2 + (y - k)

1. **Problem statement:** Graph the circle given by the equation $$(x + 1)^2 + y^2 = 81$$. 2. **Identify the center and radius:** This is the equation of a circle in standard form

1. The problem is to find the missing value in the proportion $43:14::54:x$. 2. By definition of proportion, the ratios are equal: $$\frac{43}{14} = \frac{54}{x}$$

1. The problem is to determine if the boolean expression representing the equality $4! = 4 + 0j$ is true or false. 2. Calculate $4!$ (4 factorial):

1. The problem is to understand the expression $5x^2$. 2. Here, $5x^2$ means $5$ times $x$ squared.

1. The problem is to simplify the expression $5\frac{a}{b}\sqrt{x}20$. 2. First, rewrite the expression for clarity: $5 \times \frac{a}{b} \times \sqrt{x} \times 20$.

1. **State the problem:** Solve the equation $$3 - \frac{3 - 7p}{4} = \frac{5p}{2}$$ for $p$. 2. **Rewrite the equation:** Move terms to isolate fractions and simplify.

1. **State the problem:** Compare the cost per kilogram of potatoes in London and Dublin to determine which city offers better value for money. 2. **Given information:**

1. The problem is to simplify the expression $$7^{-8} \times 7^{-7} \div 7^{-4}$$ leaving the answer with indices (exponents). 2. Use the law of indices for multiplication: $$a^m \

1. The problem is to simplify the expression $$14^{-3} \div 14^{-11} \times 14^{-19}$$ and leave the answer in indices form. 2. Recall the properties of exponents: $$a^m \div a^n =

1. State the problem: Simplify the expression $$6^{-2} \times 6^{4} \div 6^{-8}$$ leaving the answer in indices. 2. Use the laws of indices for multiplication and division: When mu

1. **State the problem:** We need to calculate the mean (average) of the given volumes: 25 cm^3, 27 cm^3, 28 cm^3, 21 cm^3, and 25 cm^3. 2. **Add all the volumes:**