🧮 algebra

1. **State the problem:** Simplify the expression $3\sqrt{50} - 5\sqrt{32} + 4\sqrt{8}$. 2. **Break down each radical:**

1. Let's start by defining rational numbers. A rational number is any number that can be expressed as the quotient or fraction $\frac{p}{q}$ of two integers, where $p$ and $q$ are

1. State the problem: Divide $$x^3 + 3x^2 - 6x - 30$$ by $$x - 3$$ and find the quotient and remainder. 2. Use polynomial long division to divide:

1. The binomial expansion of $(1 + px)^n$ starts with terms 1, $20x$, and $160x^2$. 2. The first term is always 1.

1. Solve the inequality $-2(x - 5) < 4$. Distribute to get $-2x + 10 < 4$.

1. Planteamos el problema: La empresa debe transportar azúcar blanca y rubia usando camiones tipo furgón (12 Tn) y tipo cortina (15 Tn). 2. Dado que la cantidad de camiones furgón

1. We are given that the first three terms of the expansion of $(1+px)^n$ are $1$, $20x$, and $160x^2$. 2. Using the binomial theorem, the first three terms are:

1. Solve $2(x - 3) < 4$: Expand: $2x - 6 < 4$

1. Let's start by stating the problem: Factor the quadratic expression $$x^2 + 5x + 6$$. 2. To factor a quadratic expression of the form $$x^2 + bx + c$$, we need to find two numbe

1. **State the problem:** Solve the system of simultaneous equations: $$4x + 3y = 23$$

1. The user has provided multiple function definitions and asked for compositions $g \circ f(x)$, $g \circ h(x)$, and $f \circ h(x)$ where: $$ f(x) = \frac{x}{x-1}, \quad g(x) = 3x

1. The problem states that John and Mary share a sum of money in the ratio 4:1. 2. John received 600 dollars, and we need to find Mary's share.

1. **Problem statement:** Solve the equation $9^{4/3} = 3^n$ for $n$. 2. **Rewrite the bases:** Note that $9 = 3^2$, so the equation becomes $\left(3^2\right)^{4/3} = 3^n$.

1. Stating the problem: We are given the quadratic function $$y = 4x - x^2$$ and want to analyze its graph, find key points such as x-intercepts and the vertex, and understand the

1. The question appears to ask for a redo due to an incorrect answer on question one. 2. Since the original question content is not provided here, I'll illustrate the process on a

1. Let's analyze the pattern in the given triangles one by one. 2. Each triangle has a top number and two bottom numbers side-by-side.

1. **Solve the equation:** \(\frac{m}{2} + \frac{m}{3} + 3 = 2 + \frac{m}{6}\) Step 1: Get common denominator for the fractions on the left and right. The denominators are 2,3,6; c

1. The problem is to resolve the fraction \( \frac{4}{x(x-1)} \) into partial fractions. 2. Assume the partial fraction decomposition has the form:

1. প্রশ্নটি হচ্ছে: ৭ নাম্বার গণিতে আমরা কেন $k$ এর মান ৩ ধরি? 2. চলুন দেখি, $k$ সাধারণত একটি ধ্রুবক হিসেবে ব্যবহৃত হয়। এটি কোন নির্দিষ্ট মান ধরে নেওয়ার কারণ হচ্ছে সমস্যা বা সূত্রের

1. State the problem: We are asked to decompose the expression $\frac{1}{2x^2 + x}$ into partial fractions. 2. Factor the denominator: The denominator $2x^2 + x$ can be factored as

1. Stated the problem: Calculate the sum of $\frac{8}{7}$ and $\frac{4}{5}$.\n\n2. Find a common denominator for the fractions: The denominators are 7 and 5, so the least common de