🧮 algebra

1. Problem i) Find $x$ and $y$ such that $$\begin{bmatrix} x+3 & 1 \\ -3 & 3y-4 \end{bmatrix} = \begin{bmatrix} 2 & 1 \\ -3 & 2 \end{bmatrix}$$

1. **Énoncé du problème :** Factoriser les expressions données dans l'exercice 9 et résoudre l'exercice 10. ---

1. Solve the equation $x - 15 = -15$. We want to find the value of $x$ that makes the equation true.

1. **Rounding 7.345 to one decimal place** The problem asks to round the number 7.345 to one decimal place.

1. **Problem statement:** Show that $i$ is a root of the polynomial equation $$x^4 - 5x^3 + 7x^2 - 5x + 6 = 0$$ and find the other roots.

1. **Problem statement:** Given roots $z_1=2$ and $z_2=1+\sqrt{3}i$ of the cubic equation $$z^3 + bz^2 + cz + d = 0$$ with real coefficients $b,c,d$, find: (a) the third root $z_3$

1. **Problem statement:** Given complex numbers $z = 1 + 2i$ and $w = 2 + ai$ where $a \in \mathbb{R}$, find $a$ such that:

1. **State the problem:** Solve the equation $$86x8.3^{b-1} = -1$$ for the variable $b$. 2. **Analyze the equation:** The equation involves an exponential expression with base $8.3

1. Problem: Simplify the given complex expressions and powers. 2. Recall the formula for powers of complex numbers in rectangular form: For $z = a + bi$,

1. Let's clarify the problem: you want to understand how the expression $-x - y$ is obtained. 2. The expression $-x - y$ means the negative of $x$ minus $y$. This can come from dis

1. **State the problem:** Simplify the expression $$\frac{1}{i} + \frac{1}{(2i)^3}$$ and express the answer in the form $$a + bi$$ where $$a$$ and $$b$$ are real numbers. 2. **Reca

1. The problem asks to solve question 6 by simplifying and leaving the answer in standard form $a+bi$. 2. Standard form for complex numbers is $a+bi$, where $a$ is the real part an

1. **State the problem:** Simplify the expression involving complex numbers and write the answer in standard form $a + bi$. 2. **Expression given:** $$\frac{3 + 7i \cdot (1 + 2i)}{

1. **Stating the problem:** Write the expressions \(7 \times d\) and \(d \times d\) in correct algebraic notation. 2. **Algebraic notation rules:**

1. **State the problem:** Ethan is thinking of a negative number that is lower than $-4$ and higher than $-10$. The number is odd and a multiple of 3. 2. **Identify the range:** Th

1. **State the problem:** We have a function machine where the input is 2, it is multiplied by an unknown value $x$, and the output is -10. 2. **Write the equation:** The function

1. **State the problem:** We need to find the missing number $x$ such that $$-8 \times x = 24.$$\n\n2. **Formula and rule:** To solve for $x$, we use the rule of division to isolat

1. **State the problem:** Solve the equation $$\frac{5 - x}{2x} = \frac{2x}{15 + x}$$ for $x$. 2. **Formula and rules:** To solve equations with fractions, we use cross-multiplicat

1. **Problem Statement:** We will solve the quadratic equation $$x^2 - 5x + 6 = 0$$ and justify each step. 2. **Formula and Rules:** The quadratic formula is $$x = \frac{-b \pm \sq

1. **State the problem:** Find the x-intercept and y-intercept of the linear equation $$-3x + 11y = 66$$. 2. **Recall the intercept rules:**

1. **State the problem:** Solve the system of equations: $$X + 84 = 2Y$$