🧮 algebra

1. **Problem:** Find the value of $x$ that satisfies the equation $$4^{2x+1} = 32^{x-3}$$. 2. **Formula and rules:**

1. **State the problem:** We have a balanced scale with two sides. On the left side, there are 3 identical rectangular bricks, and the total weight is 14 kg.

1. **State the problem:** We want to find the best buy for peanut butter using a 40¢-off coupon that applies to different jar sizes and quantities. 2. **Given data:**

1. **State the problem:** You want to buy stuffing mix with a 5.00 bill. There are three brands with different prices and sizes. You have a 10¢ coupon for Cook Top Stuffing Mix. Fi

1. **State the problem:** We are given the function $$f(x) = 3 + 2 \sin\left(\frac{1}{4}x\right)$$ for $$0 \leq x \leq 2\pi$$ and asked in part (d) to describe a sequence of three

1. **State the problem:** We need to describe fully a sequence of three transformations to transform the graph of $y=\sin x$ for $0 \leq x \leq \pi$ to the graph of $y=f(x)$, speci

1. **Problem Statement:** Given that the curve $y = f(|L|)$ intersects the coordinate axes exactly twice, determine which of the statements I, II, and III are necessarily true.

1. **State the problem:** Expand and simplify the expression $$(1 - 3d^4 + 4d^2)(d + 3)$$ and arrange the terms in descending powers of $d$. 2. **Recall the distributive property:*

1. **Problem:** Determine the slope of the line segment connecting points A(-2, 3) and B(3, -2). 2. **Formula:** The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is g

1. **State the problem:** Express the rational function $\frac{2x+3}{(x-1)(2x+1)(x-3)}$ as a sum of partial fractions. 2. **Formula and rules:** For distinct linear factors in the

1. **State the problem:** Simplify the expression $\sqrt{8} - \sqrt{x} \sqrt{60}$. 2. **Recall the property of square roots:** $\sqrt{a} \sqrt{b} = \sqrt{ab}$. This allows us to co

1. The problem is to simplify the expression $\sqrt{8} - \sqrt{60}$. 2. Recall that the square root of a product can be written as the product of square roots: $\sqrt{a \times b} =

1. **Stating the problem:** We need to prove the equation $$H = \frac{h}{a} \times (a + b)$$. 2. **Understanding the equation:** This equation expresses $H$ as a product of the fra

1. Nyatakan masalah: Kamisah menerima wang saku sebanyak 60 untuk $(y-6)$ hari. Setiap hari, dia membelanjakan $(x-4)$ untuk kopi dan $(x+3)$ untuk mee rebus. Kita perlu kira baki

1. The problem involves analyzing a complex inequality with nested min and max functions involving variables $x$, $y$, and $a$. 2. Since the expression is very complex and contains

1. **Problem:** Show that for any real numbers $a$ and $b$, the inequality $a^4 + b^4 > 2a^2b^2$ holds. 2. **Formula and Explanation:** We use the fact that the square of any real

1. **State the problem:** Find the value of $p(0)$ for the polynomial $p(x) = 3x^2 + 6x - 1$. 2. **Formula and explanation:** To find $p(0)$, substitute $x=0$ into the polynomial.

1. The problem is to evaluate the expression $$\frac{a - b}{\frac{1}{2}}$$ for given values of $a$ and $b$. 2. The formula used is division of a difference by a fraction:

1. **State the problem:** We are given a parabola with vertex $V(0,1)$ and directrix $x = -2$. We need to find the equation of this parabola. 2. **Recall the definition and formula

1. **Problem statement:** Given a parabola with focal length $p > 0$, show that the focal width (length of the latus rectum) is $4p$. 2. **Recall the definition:** The focal length

1. **Problem Statement:** Find the centre, vertices, minor axis points, and foci of the ellipse given by the equation $$4x^2 + 9y^2 - 16x + 18y - 11 = 0$$