🧮 algebra

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0.$$ This is a c

1. The problem asks to write the equation of the polynomial shown in Line #2, which is a cubic polynomial with a hump in the middle, indicating it has one local maximum and one loc

1. **State the problem:** We need to write the equation of the polynomial Line #2 given by $$y = a(x + 15)(x + 8)(x - 1)(x - 8)$$

1. **State the problem:** Solve the equation $$x^5 - 10x^3 + 9x = 0$$ for $x$. 2. **Formula and rules:** To solve polynomial equations, we first try to factor the expression and th

1. The problem asks to state the equation of the axis of symmetry for each function, if it exists. 2. The axis of symmetry is a vertical line that divides the graph of a function i

1. The problem asks to match each quadratic equation with its correct graph and then sketch graphs of given quadratic relations by applying transformations to the base graph $y = x

1. **State the problem:** We need to find the values of $x$ and $y$ that minimize the area of an L/T-shaped fence with a fixed perimeter of 300 meters. 2. **Identify variables and

1. **State the problem:** We start with the parabola equation $y = x^2$ and apply transformations to find the new equation in the form $y = (x - h)^2 + k$ where $h$ and $k$ represe

1. **State the problem:** Solve the equation $$\frac{x}{5} = 11 - 2x$$ for $x$. 2. **Formula and rules:** To solve for $x$, we want to isolate $x$ on one side. We can do this by el

1. **State the problem:** Solve the equation $\frac{x}{3} = 8 - x$ for $x$. 2. **Formula and rules:** To solve for $x$, we want to isolate $x$ on one side. We can do this by elimin

1. **State the problem:** We need to solve the polynomial equation $$10x^{10} + 111x^{9} - 106x^{8} - 2791x^{7} + 166x^{6} + 12469x^{5} - 774x^{4} - 2809x^{3} + 5424x^{2} - 18180x

1. **State the problem:** Solve the quadratic equation $$2x^2 - x = 6$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to set the equation to zero:

1. The problem is to analyze the function $g(x) = -\frac{1}{3}x^2 - 2x + 1$ and determine which graph correctly represents it, including identifying whether it has a maximum or min

1. **Problem:** The number of butterflies is twice as big as the number of dragonflies. How many butterflies are sitting on the two blossoms? 2. **Formula and rules:** Let the numb

1. **State the problem:** Solve the equation $4(2x-1) = 3(x+2)$. 2. **Apply the distributive property:** Multiply each term inside the parentheses by the factor outside.

1. The problem is to graph the system of equations: $$y = x + 1$$

1. **State the problem:** We need to graph the system of equations and find the point where the two lines intersect. 2. **Write down the system:**

1. **State the problem:** We are given two linear equations: $$y = 2x - 4$$

1. **State the problem:** We need to find the solution to the system of equations: $$y = -2x + 7$$

1. **State the problem:** Divide the polynomial $$2x^4 - x^3 - 8x^2 + 15x - 6$$ by the binomial $$x + 4$$. 2. **Recall the division formula:** Polynomial division is similar to lon

1. **State the problem:** Simplify or evaluate the expression $$7) \log_c \frac{a+1}{b^8}$$. 2. **Recall the logarithm rule:** The logarithm of a quotient is the difference of the