🧮 algebra

1. **Problem statement:** (a)(i) Find the angle from line $l_1: y = -5x + 7$ to line $l_2: y = 3x - 8$.

1. The problem is to solve the quadratic equation $x^2 - 6x - 3 = 0$. 2. We use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a=1$, $b=-6$, and $c=-3$.

1. **State the problem:** Factor the quadratic expression $x^2 - 4$. 2. **Recognize the form:** The expression is a difference of squares since $x^2$ and $4$ are both perfect squar

1. **Statement of the problem:** We study the functions \(\psi(x) = -x^6 + 6x^2 - 2\) and \(\varphi(x) = x^4 - 4x + 6\) defined on \(\mathbb{R}\). We analyze their limits, monotoni

1. **Stating the problem:** We are given a lattice $(L, \leq)$ and elements $a, b \in L$. We want to understand and prove the equivalence: $$a \leq b \iff a \wedge b = a \iff a \ve

1. **State the problem:** We are given the total monthly profit $P = 250000$, and the monthly revenue $R$ is twice the monthly cost $C$ minus 60000. We need to find $R$ and $C$.

1. **Problem 1.1**: Given the quadratic function $$f(x) = -x^2 - 2x - 1$$ 1. Find the x- and y-intercepts.

1. Problem: An alloy contains 120 g of copper and 150 g of zinc. (i) Find the ratio of copper to zinc.

1. **State the problem:** Simplify the expression $$(2x^2 - 3x - 6) - (-x^3 + 4x^2 - 5)$$. 2. **Remove parentheses carefully:** When subtracting a polynomial, change the signs of e

1. The problem is to compare the two fractions $\frac{4}{17}$ and $\frac{7}{10}$. 2. To compare fractions, we can find a common denominator or convert them to decimals.

1. The problem is to evaluate the expression $3 \div 2 \times \left( \frac{1}{6} \times 3.14 \times 2^3 \right)$.\n\n2. First, calculate the exponent: $2^3 = 8$.\n\n3. Next, multip

1. **State the problem:** Solve for $x$ in the equation $$\frac{29}{36}x + 120 = x.$$\n\n2. **Isolate the variable terms:** Subtract $\frac{29}{36}x$ from both sides to get all $x$

1. **State the problem:** We need to solve the expression $$\frac{29}{36} + 120$$. 2. **Convert the whole number to a fraction:** To add a fraction and a whole number, express the

1. **Solve the system:** $$\frac{2}{x} + \frac{2}{3y} = \frac{1}{6}, \quad \frac{3}{x} + \frac{2}{y} = 0$$

1. The problem involves simplifying the expression $$\frac{1}{4} \times \frac{5}{9} \times 120$$. 2. First, multiply the numerators and denominators of the fractions: $$\frac{1 \ti

1. The problem asks to find the result of the integral, but the expressions given are unclear as written. Let's interpret the left and right columns as separate expressions to simp

1. **State the problem:** We are given two equations: $$x^2 - y^2 = 30$$

1. To answer "Where did the six come from?", I need to know the specific problem or equation you are referring to. 2. The number six could appear as a coefficient, constant, or res

1. **State the problem:** We are given two equations: $$x^2 - y^2 = 30$$

1. **State the problem:** We are given two equations: $$x^2 - y^2 = 30$$

1. **State the problem:** Expand and simplify the expression $$(3p - 7)(4p - 8)$$. 2. **Apply the distributive property (FOIL method):** Multiply each term in the first parenthesis