🧮 algebra

1. **State the problem:** Find the vertex and evaluate the quadratic function $y=3x^2 - 30x + 67$ at specific points. 2. **Formula for vertex:** The vertex $x$-coordinate of a para

1. **State the problem:** Leona answered all 40 questions on a test and got 90% of them correct. We need to find how many questions she answered correctly. 2. **Formula used:** To

1. The problem involves understanding the ratio of vinegar to oil given the quantities 60 units of vinegar and 240 units of oil. 2. The ratio of vinegar to oil is calculated by div

1. **State the problem:** We are given the quadratic function $$y = (x - 1)(x - 3)$$ and asked to find the coordinates of the x-intercepts.

1. **State the problem:** Find the vertex of the quadratic function $$y = -3x^2 + 4x - 2$$. 2. **Recall the vertex form:** A quadratic function can be written as $$y = a(x-h)^2 + k

1. **State the problem:** We are given the quadratic expression $2x^2 - 11x + 9$ and want to analyze its graph and find the minimum point $P$. 2. **Rewrite the quadratic in vertex

1. **State the problem:** The carpenter builds 11 bookshelves and 25 tables. Each bookshelf requires 1 box of screws, 4 pieces of 2 x 4's, and 4 sheets of plywood. Each table requi

1. The problem is to find the vertex of each quadratic function given. 2. The vertex form of a quadratic function is $$y = a(x-h)^2 + k$$ where $(h,k)$ is the vertex.

1. **State the problem:** Simplify the expression $$(2x + 1)^0$$. 2. **Recall the rule:** Any non-zero number or expression raised to the power of zero equals 1. That is, for any $

1. **State the problem:** Solve the equation $$7 - 2 \cdot (3y + 4) = 5 - 4y$$ for $y$. 2. **Apply the distributive property:** Multiply $-2$ by each term inside the parentheses.

1. **State the problem:** Simplify the expression $$\frac{5 - v}{2 - 2\cdot(v+1)} \div \frac{1 - v}{\frac{3}{2}}$$. 2. **Rewrite the division as multiplication by the reciprocal:**

1. **Énoncé du problème** : Trouver pour chaque parabole les points d'intersection avec les axes, l'axe de symétrie et le sommet. ---

1. Énoncé du problème : Refaites l'étude du signe d'une expression algébrique donnée. 2. Rappel de la méthode : Pour étudier le signe d'une expression, on identifie les racines (va

1. Énoncé du problème : Déterminer les variations de la fonction $f$ telle que $u_n = f(n)$ pour la suite $u_n = 3n^2 - 4$ et en déduire les variations de $u$. 2. Formule et règles

1. The problem asks to evaluate $2^{-3}$. 2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$ where $a \neq 0$ and $n$ is a positive integer.

1. **Problem statement:** Löse die Gleichung $$5 \cdot (x + 3) = 2 \cdot x + 12 + 3 \cdot x$$. 2. **Formel und Regeln:** Um lineare Gleichungen zu lösen, verwenden wir die Distribu

1. **State the problem:** We are given the quadratic function $f(x) = -2x^2 - 12x - 7$ and need to analyze its properties. 2. **Formula and rules:** A quadratic function is general

1. The problem is to find linear equations that pass through the points where the stars are located: (2,8), (2,4), (8,7), and (14,4). 2. Recall that a linear equation in two variab

1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} x + 1, & x \leq -2 \\ -6, & -2 < x \leq 2 \\ 5, & x > 2 \end{cases}$$

1. **State the problem:** We are given the quadratic function $$h(x) = -\frac{1}{2}x^2 + 4x - 10$$ and need to find its vertex, line of symmetry, x-intercepts, and y-intercept. 2.

1. **State the problem:** We are given a piecewise function: $$f(x) = \begin{cases} -x - 4, & x < -3 \\