🧮 algebra

1. The problem gives a system of two equations: $$-8x - 6y = 44$$

1. **State the problem:** Simplify the expression $$(3-x)^2 (4+x)^2$$. 2. **Rewrite the expression:** Notice this is a product of two squared binomials:

1. **State the problem:** Multiply the two polynomials $$(x^2 + 5x + 4)(x^2 + 2x - 1)$$.

1. **Problem Statement:** Given the quadratic equation $$2x^2 - 3x - 1 = 0$$ with roots \(\alpha\) and \(\beta\), find new equations with integer coefficients whose roots are: (a)

1. We are asked to solve the equation $$1^n + 2^n + 3^n + 4^n = 7^{n-1} + 11$$ for $n$. 2. Notice that $1^n = 1$ for any integer $n$, so the left side simplifies to $$1 + 2^n + 3^n

1. Evaluate $4 + (-2) \times (-2)$. Calculate the product first: $(-2) \times (-2) = 4$.

1. **Problem:** Prove that $n^2 - 7n + 12 \geq 0$ for all positive integers $n \geq 2$ using mathematical induction. 2. **Step 1: Base case** ($n=2$):

1. The problem asks to fill in the blanks for multiplication and division with positive and negative numbers. 2. a) Calculate $4 \times (-6)$.

1. **Stating the problem:** Simplify each expression by applying the laws of exponents in parts (c), (d), and (e). 2. **Part (c): Simplify \((u^{-1} v / v^{-1})^2\)**

1. Énoncé du problème : Trouver les solutions de l'équation \(-2x^2 + x + 1 = 0\). 2. Identifier les coefficients : \(a = -2\), \(b = 1\), \(c = 1\).

1. The problem involves expressions listed in three different boxes with algebraic terms. 2. The expressions in the center box are:

1. The problem shows several matrix and algebraic expressions to simplify or combine. 2. For each matrix or expression, interpret and simplify step-by-step.

1. **Stating the problem:** Uranium decomposes at a rate proportional to the present amount. Initially, there are 55 grams, and after 18 years, 0.75% of the original amount has dec

1. Given $a^3 + \frac{1}{a^3} = 18$, find $a^2 + \frac{1}{a^2}$. Step 1: Recall the identity for cubes:\

1. Solve the quadratic equation $x^2 - 2x + 5 = 0$. Use the discriminant $\Delta = b^2 - 4ac = (-2)^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16$. Since $\Delta < 0$, the roots are complex

1. The problem asks to find the distances for the intervals 0-5, 5-10, 10-15, 15-20, and 20-25. 2. Since no specific function or context was provided in the previous message, I ass

1. State the problem: We are given two bags of chips, one 12 ounces costing 3.96 and another 16 ounces costing 5.28. We want to find which bag is the better deal by comparing the c

1. Let's clarify the problem: You asked which graph corresponds to the options you gave, but no options or functions were provided in your message. 2. To help you identify the corr

1. Let's start by understanding what power means in mathematics. 2. Power refers to exponentiation, which means multiplying a number by itself a certain number of times.

1. **Stating the problem:** We are given the function $$y = \frac{1}{x} + 5$$ and asked to understand and explain its graphs and asymptotes. 2. **Analyze the function:** The functi

1. Problema: Calcular $$F = 4^{20} - 16^{10}$$. Sabemos que $$16 = 4^2$$, entonces