🧮 algebra

1. **Problem statement:** Given that $y$ varies directly with $(x-1)^2$, and when $x=4$, $y=18$, find: i. The relationship between $x$ and $y$

1. Problem: Check for mistakes in the given algebraic expansions and factorizations. 2. We will verify each item by expanding or factoring and comparing with the provided expressio

1. **State the problem:** We have a piecewise function $$g(x) = \begin{cases} x - 2 & x \leq 3 \\ x + 1 & 3 \leq x \leq 5 \end{cases}$$

1. The problem is to understand why we multiply by the conjugate of the numerator instead of the denominator when simplifying expressions involving radicals. 2. The conjugate of a

1. **State the problem:** Solve the simultaneous equations: $$2x + 3y = 6$$

1. **Problem statement:** Determine if the function $f(x) = \sqrt{x} + x$ is injective. 2. **Definition:** A function is injective (one-to-one) if for every $x_1, x_2$ in the domai

1. **State the problem:** Solve the simultaneous equations: $$2xy=45$$

1. **State the problem:** We need to find an angle $x$ such that the sum of its complementary and supplementary angles is 200 degrees. 2. **Recall definitions:**

1. **Problem Statement:** Find the sum of the largest and smallest 5-digit numbers formed by the digits 2, 0, 6, 1, 9. 2. **Important Rule:** A 5-digit number cannot start with 0 b

1. **Problem:** Given points (-2,-6), (0,2), (2,5), (3,9), (4,26), (5,35) representing (x,y), find constants $a$ and $b$ such that $y = a \times x + b$ fits the data. 2. **Formula:

1. The problem is to simplify the expression $$\frac{ab + b}{b}$$ and verify if it equals $$a$$. 2. The formula used here is the property of division over addition: $$\frac{x + y}{

1. **Problem statement:** A football team plays 30 matches. Each win gives 5 points, each draw gives 2 points, and each loss deducts 1 point. The team scored a total of 108 points.

1. **State the problem:** Find the sum of the largest and smallest 5-digit numbers formed using the digits 3, 0, 4, 1, 9. 2. **Important rule:** A 5-digit number cannot start with

1. **Problem statement:** Given the quadratic equation $$12a^2 - 30a - 1 = 0,$$ find the value of $$4a^2 + \frac{1}{36a^2}.$$\n\n2. **Step 1: Understand the problem and what is ask

1. **State the problem:** Evaluate the expression $$\left(\frac{1}{3} + \frac{2}{9}\right)^9 + \frac{\left(\frac{3}{4}\right)^2 / 9}{\frac{9}{4} \times (-4)}$$ and determine which

1. **State the problem:** Solve the equation $$Z^{5} + Z^{3} + Z = 0$$ for $Z$. 2. **Use the factoring technique:** Notice that each term contains a factor of $Z$, so factor it out

1. The problem asks for the function of $g^n(x)$ in terms of $n$ and $x$. 2. Here, $g^n(x)$ denotes the $n$-th iterate of the function $g$, meaning applying $g$ to $x$ repeatedly $

1. The problem asks to find $g^2(x)$ when $g(x) = x - 3$. 2. The notation $g^2(x)$ means the function $g$ composed with itself: $g(g(x))$.

1. The problem is to understand and describe the graph of the function $y = x^x$. 2. The function $y = x^x$ is defined for $x > 0$ because for real numbers, raising a negative numb

1. **State the problem:** Simplify the expression $$\frac{x^2 - 4x}{x^2 - 9}$$. 2. **Recall the formulas and rules:**

1. State the problem: Compute $1\sqrt{126}$.\n2. Formula and rules: Use the product rule for square roots $\sqrt{ab}=\sqrt{a}\sqrt{b}$.\n3. Factor 126 into primes: $126=2\cdot3^2\c