🧮 algebra

1. **State the problem:** Find all values of $m$ for which the equation $$4 = -(m - 1)w^2 - 8w$$ has exactly one real solution in $w$. 2. **Rewrite the equation:** Move all terms t

1. **State the problem:** Find all values of $n$ for which the quadratic equation $$(n + 1)s^2 - 4s - 6 = 0$$ has exactly one real solution. 2. **Recall the condition for one real

1. The problem is to simplify the expression $y = \frac{1}{3} - \frac{2}{7} \times \frac{5}{4}$. 2. According to the order of operations, multiplication must be done before subtrac

1. The problem is to make $c$ the subject of the formula from $f = 5c - 8$. 2. Start with the equation:

1. **State the problem:** A customer bought a pumpkin weighing 4.8 kilograms (rounded to the nearest tenth) and paid 1.38 per pound. We need to find which total amount paid is poss

1. **Énoncé du problème :** Montrer que $3a^2 + 8\sqrt{3} a \geq -16$. 2. **Formule et règles importantes :**

1. **Stating the problem:** We have a new sequence formed by taking the natural numbers and repeating every number divisible by 3 once more immediately after it. For example, the f

1. **بيان المسألة:** لدينا المتتالية العددية $\left(u_n\right)$ المعرفة ب: $$u_{n+1} = \frac{3u_n + 2}{2 + u_n}, \quad u_0 = \frac{3}{2}$$

1. **State the problem:** Solve the equation $$2x + \frac{138x^3 - 192x^2 + 126x + 2}{(x-1)^4} = 0.$$\n\n2. **Rewrite the equation:** Multiply both sides by $(x-1)^4$ to clear the

1. Planteamos el problema: calcular $$m=\sqrt[\frac{200-\pi}{\pi}]{\frac{180}{\pi}}$$ donde la raíz tiene índice $$\frac{200-\pi}{\pi}$$ y el radicando es $$\frac{180}{\pi}$$. 2. R

1. **Problem statement:** Factorise the quadratic expressions given. 2. **Formula and rules:** To factorise a quadratic expression of the form $x^2 + bx + c$, find two numbers that

1. **Problem statement:** Factorise the quadratic expression $x^2 + 5x + 6$. 2. **Formula and rules:** To factorise a quadratic expression of the form $ax^2 + bx + c$, we look for

1. **State the problem:** Solve the equation $$2x + \frac{(x-1)^2 (128x) - (64x^2 + 2x - 2)}{(x-1)^4} = 0$$ for $x$. 2. **Rewrite the equation:** To combine terms, write the first

1. **State the problem:** Solve the equation $$2x + \frac{(x-1)^2 (128x) - (64x^2 + 2x - 1)}{(x-1)^4} = 0.$$\n\n2. **Rewrite the equation:** To combine terms, write $$2x$$ as $$\fr

1. **State the problem:** Solve the linear equation $2x + 5 = 0$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate $x$ by performing inverse operations. Subtract 5 from

1. **State the problem:** We need to graph the quadratic function $f(x) = x^2 + 4x + 1$ and find the coordinates of its minimum point. 2. **Recall the formula for the vertex of a p

1. **State the problem:** We need to find the equation of the line passing through points such as $(-9,-9)$, $(-6,-6)$, $(0,0)$, $(3,3)$, $(6,6)$, $(9,9)$, and $(12,12)$ in slope-i

1. **Énoncé du problème :** Vérifier que le polynôme $P(X) = X^7 + X^6 - X^3$ peut s'écrire sous la forme

1. The problem asks us to interpret the horizontal intercept (85,0) on a graph showing combinations of bananas and apples sold. 2. The horizontal intercept is the point where the g

1. **State the problem:** We want to find how many times the goal distance of 4 \frac{1}{2} miles fits into the actual distance biked, which is 6 miles.

1. The problem is to rewrite the expression $$\left(m^{\frac{2}{3}} n^{-\frac{1}{3}}\right)^{6}$$ using only positive integer exponents. 2. We use the power of a product rule: $$\l