🧮 algebra

1. **State the problem:** We analyze the parabola given by the equation $y = (x + 2)^2 + 3$. 2. **Identify the vertex:** The vertex form of a parabola is $y = a(x - h)^2 + k$, wher

1. **Stating the problem:** We are given a table of water temperatures over time and asked to find a quadratic function $f(x) = ax^2 + bx + c$ that models the temperature, where $x

1. The problem asks to identify the two inequalities that form the system graphed. 2. The graph shows a dashed upward-opening parabola with vertex at (1,0) and a solid decreasing l

1. **Problem statement:** We have a theatre seating arrangement where row 1 has 28 seats, and each subsequent row has one more seat than the previous row.

1. **State the problem:** Solve the quadratic equation $2x^2 - 10 = 26$ for $x$. 2. **Write the formula and rules:** To solve quadratic equations, we isolate the $x^2$ term and the

1. **State the problem:** Solve the inequality $$t - 1(t - 6)(t + 1) < 0$$. 2. **Rewrite the inequality:** The expression is ambiguous, so we interpret it as $$t - 1 \cdot (t - 6)(

1. The problem asks to find the coordinate point that represents the solution to a system of linear equations graphed on a coordinate plane. 2. The solution to a system of linear e

1. **State the problem:** Find the value of $z$ that satisfies the equation $$\frac{1}{4}(2 - z) + \frac{3}{2} = 3(z - 2).$$ 2. **Write the equation:** $$\frac{1}{4}(2 - z) + \frac

1. **State the problem:** Solve the equation $$(2 - 3\ln)^{\frac{4}{3}} = 256$$ for $\ln$. 2. **Rewrite the equation:** Recognize that 256 is a power of 4, since $$256 = 4^4$$.

1. **State the problem:** We are given the function $$y = 1 + 3\sqrt{x + 3}$$ and asked to graph it along with its inverse. 2. **Recall the formula and rules:** The function involv

1. The problem is to graph the inverse of a given function. 2. To find the inverse of a function $y=f(x)$, we swap $x$ and $y$ and solve for $y$.

1. **Problem 3:** Find the inverse of the function $g(x) = -3 + (x - 1)^3$ and graph both the function and its inverse. 2. **Step 1:** Write the function as $y = -3 + (x - 1)^3$.

1. **Problem 2:** Simplify the expression $$\frac{\sqrt[3]{3m^{2}n^{4}}}{\sqrt[3]{4m^{4}n^{4}}}$$. 2. Use the property of cube roots: $$\frac{\sqrt[3]{a}}{\sqrt[3]{b}} = \sqrt[3]{\

1. **State the problem:** Solve the inequality $$x^4 + x^3 - x^2 - x > 0$$ and write the solution in interval form. 2. **Factor the expression:** Group terms to factor by grouping:

1. **State the problem:** Find the inverse of the function $$g(x) = -3 + (x-1)^3$$. 2. **Recall the formula for inverse functions:** To find the inverse function $$g^{-1}(x)$$, we

1. **State the problem:** Solve for $x$ in the equation $$-2(x - 3) = 3x + 41$$. 2. **Apply the distributive property:** Multiply $-2$ by each term inside the parentheses:

1. The problem asks to show that the coordinates of point C are (4, 0). 2. We are given the equation of the line in point-slope form: $$y - y' = m(x - x')$$ where $m$ is the slope

1. **State the problem:** We are given points A(2, 3), B(1, -2), and point C unknown. The angle between lines AB and BC is 45° (acute). We need to find the slope of AB and then use

1. **State the problem:** Solve for $r$ in the equation $r - 14.2 = 1.89$. 2. **Formula and rules:** To isolate $r$, add $14.2$ to both sides of the equation to cancel out the $-14

1. **State the problem:** Solve for $j$ in the equation $$3j = 7.83$$. 2. **Formula and rules:** To isolate $j$, divide both sides of the equation by 3. Remember, dividing both sid

1. Factor the polynomial $wp + 2n + 8p + 16$. 2. Factor the polynomial $3bc - 2b - 10 + 15c$.