🧮 algebra

1. **State the problem:** We want to find the value of $k$ such that the piecewise function $$f(x) = \begin{cases} k^3 + x & \text{if } x < 3 \\ \frac{16}{k^2 - x} & \text{if } x \

1. The problem is to find integer solutions to an equation or system (please specify the exact equation if needed). 2. To find integer solutions, we typically look for values of va

1. **State the problem:** Solve the equation $$104 = 8y^2 + 8$$ for $y$. 2. **Isolate the term with $y^2$:** Subtract 8 from both sides:

1. Stating the problem: Simplify the expression $\frac{a}{b}\left(\frac{c}{d} + \frac{e}{f}\right)$. 2. Use the distributive property and fraction addition rules:

1. **State the problem:** Find the average rate of change of the function $g(x) = 2x^3 + 2$ from $x = -2$ to $x = 3$. 2. **Formula:** The average rate of change of a function $g(x)

1. **State the problem:** Simplify the expression $\sqrt{9x^2+6x-3}$. 2. **Identify the expression inside the square root:** The radicand is $9x^2 + 6x - 3$.

1. **State the problem:** Solve the system of equations using substitution: $$4x + 3y = 37$$

1. **State the problem:** Factor the expression $$x^2 + 2x + 1 - y^2$$. 2. **Recognize the structure:** The expression can be seen as a difference of squares because $$x^2 + 2x + 1

1. **State the problem:** We are given that the area of Manitoba is about $1 \frac{1}{5}$ times the total area of the four Atlantic provinces, and the area of Yukon Territory is ab

1. **State the problem:** Calculate the product of $2 \frac{3}{4}$ and $-4 \frac{2}{3}$. 2. **Convert mixed numbers to improper fractions:**

1. **State the problem:** Find the zeros (roots) of the quadratic equation $$5x^2 + 19x + 12 = 0$$. 2. **Formula and approach:** To find the roots, we can factor the quadratic if p

1. **State the problem:** Evaluate the expression $$4x^2 - by^2 - by$$ for $$x=2$$, $$b=\frac{1}{2}$$, and $$y=-3$$. 2. **Write the expression with substituted values:**

1. The first problem asks to sketch the function $f(x) = (x - 5)^2$. 2. This is a parabola that opens upwards with its vertex at the point $(5, 0)$.

1. **Problem statement:** We want to find the correct equation for the rocket's weight $W$ at time $t$ where $0 \leq t \leq T$, given initial weight $W_1$ and final weight $W_T$ at

1. **State the problem:** The ratio of female to male employees is initially 2:5. After hiring 6 more women, the ratio becomes 7:10. We need to find the total number of employees a

1. **State the problem:** We have a hospital with two wards, A and B. Ward A has 40% of beds and 60% of turnover of 120 million. Ward B has 60% of beds and 40% of turnover. The bed

1. **Problem statement:** Claudia spends $x$ percent of her monthly work time on the project "Customer Survey" for the first $a$ months, and $y$ percent for the remaining $b$ month

1. **State the problem:** 8 employees complete the inventory in 6 days. We want to find how long it takes if 4 more employees join, making 12 employees. 2. **Formula used:** Work d

1. **State the problem:** We have a box formed by cutting out squares of side $x$ cm from each corner of a 9 cm by 9 cm square sheet and folding up the sides. We want to express th

1. **State the problem:** We have a box formed by cutting out squares of side $x$ cm from each corner of a $9 \times 9$ cm square sheet and folding up the sides. We want to express

1. **State the problem:** Find the equation of the line passing through the points $(-3, h(-3))$ and $(1, h(1))$ where $h(x) = x^3 + 2$. 2. **Calculate the points:**