🧮 algebra

1. **Problem statement:** Find the coordinates of point A on the parabola $y = ax^2$ where an equilateral triangle $OAB$ with side length 4 lies, with $O$ at the origin.

1. The problem is to find which area model represents the expression $ (2x + 3)(3x + 3) $. 2. The formula to expand this product is the distributive property: $$ (a + b)(c + d) = a

1. **Stating the problem:** We have an area model with tiles arranged in 3 rows and 5 columns representing algebraic terms. The first two rows have 3 blue squares labeled $x^2$ and

1. **Problem statement:** Find the lengths of line segments $OA$ and $OB$ where points $A$ and $B$ are the intersections of the line $y=x$ with the parabolas $y=x^2$ and $y=-\frac{

1. **State the problem:** Simplify and solve the equation $2i + x + 3y = -7i^{15}y - 5x$ for $x$ and $y$. 2. **Recall important rules:**

1. **State the problem:** Simplify the expression $\frac{1}{i^{50}}$ where $i$ is the imaginary unit with the property $i^2 = -1$. 2. **Recall the powers of $i$:** The powers of $i

1. The problem is to simplify the expression $\frac{\sqrt{5}}{\sqrt{1}}$. 2. Recall the property of square roots: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$.

1. **State the problem:** Solve the system of linear equations: $$5x - y = -5$$

1. **Stating the problem:** We need to find the first term of the expression $\left(\frac{16}{9} \times \left(-\frac{3}{2}\right)\right)^6$. 2. **Understanding the expression:** Th

1. **State the problem:** Solve the system of equations using substitution: $$y = x + 6$$

1. **Problem Statement:** Given the graph of a rational function $f$ with vertical and horizontal asymptotes and intercepts, find:

1. **State the problem:** Factor the polynomial $4x^3 - 26x^2 + 30x$ completely. 2. **Identify the common factor:** Each term has a factor of $2x$, so factor it out first.

1. **State the problem:** Find the area of the shaded region, which is the area of the larger square minus the area of the smaller square cut out from inside it.

1. **Problem:** Transform the parabola equation from $x^2 = 2y + 11$ to $2(x - 1)^2 = 2y + 5$ and verify if the point $(-5,7)$ lies on the new parabola. 2. **Step 1: Understand the

1. **State the problem:** Simplify the expression $\sqrt{27} \times \sqrt{80}$.\n\n2. **Recall the property of square roots:** $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$. This

1. **State the problem:** Simplify the expression $\sqrt{450}$. 2. **Recall the formula and rules:** The square root of a product can be expressed as the product of the square root

1. **State the problem:** Find the middle zero of the function $f(x) = x^5 (x - 1)^2 (x + 8)$ and its multiplicity. 2. **Identify zeros:** The zeros are $x = -8$, $x = 0$, and $x =

1. **State the problem:** Find the smallest zero of the function $f(x) = x^5 (x - 1)^2 (x + 8)$ and its multiplicity. 2. **Identify zeros:** The zeros are the values of $x$ that ma

1. **Stating the problem:** A birdwatcher sees an average of $p$ penguins each day. We want to find how many penguins they would expect to see in one hour. 2. **Understanding the p

1. The problem asks for the range of the function $g(x) = x^2$. 2. The function $g(x) = x^2$ is a quadratic function that squares any real number $x$.

1. **State the problem:** Simplify the expression $$h^3 \times c^3 \times h c^4$$ and write the answer with only positive exponents. 2. **Recall the exponent rules:**