🧮 algebra

1. **State the problem:** Simplify the expression $$\frac{3c^2 - 7c + 4}{9c^2 - 16}$$. 2. **Factor the numerator:** We look for two numbers that multiply to $3 \times 4 = 12$ and a

1. **State the problem:** Given the quadratic function $g(x) = 3x^2 - 8x + 2$, (i) write $g(x)$ in the form $a(x+h)^2 + k$ (vertex form).

1. **State the problem:** Calculate the following expressions involving Fibonacci numbers $F_n$:

1. **Problem Statement:** Complete the table for the function $y = x^2 - 2x + 2$ at given $x$ values. 2. **Formula:** Use the function $y = x^2 - 2x + 2$ to calculate $y$ for each

1. **State the problem:** Simplify the expression $$\frac{w^2 + w - 56}{w + 8}$$. 2. **Recall the formula and rules:** To simplify a rational expression, factor the numerator and d

1. **State the problem:** We have an election with 4 candidates (A, B, C, D) and voters' preferences distributed as given. We need to find (a) the winner of the election and (b) th

1. **State the problem:** Simplify the expression $$- \{3 + [-(-9) \cdot 4 - 6 \div 3]\} + 5 \cdot 2 \cdot [7 - 10] \div 3$$. 2. **Recall order of operations:** Use PEMDAS (Parenth

1. **State the problem:** We need to complete the table for the function $y=3^x$ at $x=-1,0,1,2$ and then plot two points to graph the function. 2. **Recall the function:** The fun

1. **State the problem:** Evaluate each expression given the values of $x$. 2. **Given values:**

1. **State the problem:** Evaluate the expression $4 \cdot 61 - 3x$ when $x = \frac{1}{3}$. 2. **Substitute the value of $x$:** Replace $x$ with $\frac{1}{3}$ in the expression:

1. **State the problem:** Find the remainder when the polynomial $$f(x) = 2x^5 - 5x^2 + 2$$ is divided by $$x + 3$$. 2. **Recall the Remainder Theorem:** When a polynomial $$f(x)$$

1. The problem asks to simplify the fraction $\frac{120}{500}$. 2. To simplify a fraction, we divide the numerator and denominator by their greatest common divisor (GCD).

1. Stwierdźmy problem: Rozwiążemy równanie $$\frac{1}{3}(6x - 12) + \frac{x}{2} = \frac{1}{5}(x + 3)$$ oraz obliczymy wartość prawej strony dla znalezionej wartości $x$. 2. Zastosu

1. Stwierdźmy, które z podanych równań jest sprzeczne, czyli nie ma rozwiązania. 2. Rozwiążmy każde równanie po kolei.

1. **Stating the problem:** We have a triangle with sides given by expressions $20 - 2x$, $\frac{1}{2}x + 10$, and $x^2 - x$. We need to find the value of $x$ for which the triangl

1. Stated problem: Simplify the expression $$4(11x + 4y) + 5(2x + 4y)$$ and find which given option is equivalent. 2. Use distributive property: $$a(b + c) = ab + ac$$.

1. **State the problem:** Solve the system of equations using elimination: $$\begin{cases} x - 2y = 10 \\ 3x + y = -12 \end{cases}$$

1. **State the problem:** Simplify the expression $$\frac{x^2 - 49}{x^2 + 2x - 35} \times \frac{2x^3 - 18x^2 + 40x}{3x^2 - 21x}$$ and find the restrictions on the variable $x$. 2.

1. **Problem a:** Simplify the expression \( \frac{x^2 - 49}{x^2 + 2x - 35} \times \frac{2x^2 - 18x^2 + 40x}{3x^2 - 21x} \). 2. **Step 1:** Factor all polynomials.

1. **State the problem:** Solve for $r$ in the equation $$10 P_{n,r} = 10 P_{n,r+1}$$ where $P_{n,r}$ is the number of permutations of $n$ items taken $r$ at a time. 2. **Recall th

1. **State the problem:** We need to find the equations of lines perpendicular to given lines (Line 4, Line 5, Line 6) passing through specific points: (3, 8), (-12, 6), and (-7, -