🧮 algebra

1. **State the problem:** We need to find the value of $m$ such that $$x^3 - 8 + m = (x + 1)(x^2 - x + 1).$$ 2. **Recall the formula:** The right side is a product of a binomial an

1. **Problem 16: Solve the system of inequalities by graphing:** Given inequalities:

1. Let's analyze the expression where $a$ is raised to the power 2, i.e., $a^2$. 2. The key property of exponents is that squaring any real number, whether positive or negative, re

1. **State the problem:** Solve for $x$ in the equation $$\frac{3}{4} = \frac{12}{x}$$. 2. **Formula and rule:** When two fractions are equal, their cross products are equal. This

1. Let's clarify the problem: You are asking why the answer is not simply positive or negative 3, as in option (d). 2. Typically, when solving equations like $x^2 = 9$, the solutio

1. **State the problem:** Given that $a^2 - b^2 = 45$ and $a - b = 5$, find the value of $\sqrt{a + b}$. 2. **Recall the formula:** The expression $a^2 - b^2$ can be factored using

1. **State the problem:** Solve for $x$ in the equation $$\frac{8}{3} = \frac{-x - 9}{7x + 4}.$$\n\n2. **Use the cross-multiplication rule:** When two fractions are equal, their cr

1. **State the problem:** Find the magnitude of the complex number $z = -7 + 3i$. 2. **Formula:** The magnitude (or modulus) of a complex number $z = a + bi$ is given by:

1. We are asked to calculate $ (2 - 3i)^5 $ and express the result in the form $ a + bi $. 2. To solve this, we can use the binomial theorem for powers of complex numbers:

1. **State the problem:** Simplify the expression $$6 \left( \frac{11a}{35} + \frac{b}{35} \right) - \left( \frac{9a}{35} - \frac{27b}{35} + \frac{20a}{35} \right) + \left( \frac{a

1. **State the problem:** Solve for $x$ in the equation $$2 + 5 \ln(7x - 1) = 17.$$\n\n2. **Isolate the logarithmic term:** Subtract 2 from both sides to get $$5 \ln(7x - 1) = 15.$

1. The problem is to understand and find the vertical asymptote of a function. 2. A vertical asymptote occurs where the function approaches infinity or negative infinity as the inp

1. **State the problem:** We are given the function $y = 7x^2 - 4$ and want to analyze it. 2. **Formula and rules:** This is a quadratic function of the form $y = ax^2 + bx + c$ wh

1. State the problem: Expand and simplify the expression $(x-7)(x+3)$. 2. Formula and rule: Use the distributive property (FOIL) which expands products of binomials.

1. **State the problem:** Simplify and factor the quadratic expression $x^2 - 4x - 21$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two number

1. **State the problem:** We are given the midpoint $M(3,4)$ of points $A(2,a)$ and $B(b,12)$, and we need to find the values of $a$ and $b$. 2. **Formula for midpoint:** The midpo

1. Énonçons le problème : calculer l'expression $2\sqrt{7} \div 5 - \sqrt{15}$.\n\n2. Rappelons la formule et les règles importantes : la division et la soustraction doivent être e

1. **State the problem:** Simplify the expression $2\sqrt{7} \div 5 - \sqrt{15}$. 2. **Rewrite the division:** Division by 5 can be written as multiplication by $\frac{1}{5}$, so t

1. The problem asks for the key to determine the next draw, which typically refers to finding a pattern or rule to predict the next number or outcome in a sequence or draw. 2. To s

1. The problem involves simplifying and understanding the sum of a geometric series given by $$\sum_{j=0}^{r-1} (a^{-i})^j = \frac{(a^{-i})^r - 1}{a^{-i} - 1}$$

1. **State the problem:** We need to find the quadratic equation $y = ax^2 + bx + c$ given that the graph has a turning point at $(-2,4)$ and passes through the point $(0,8)$. 2. *