🧮 algebra

1. Problem: Löse die Gleichung a) 6 + 3 \cdot (12 + x) = 54 und mache die Probe. 2. Formel und Regeln: Wir lösen lineare Gleichungen der Form $a + b(x + c) = d$.

1. **Problem (c): Simplify** $\frac{3 \sqrt{22}}{\sqrt{11}}$. 2. **Formula and rules:** To simplify a fraction with square roots, use the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqr

1. **State the problem:** We are given two quadratic functions: - $y = -2(x - 1)^2 + 3$

1. **State the problem:** Find the points of intersection between the quadratic function $$y = x^2 - 5x - 2$$ and the linear function $$y = -x + 3$$. 2. **Set the equations equal t

1. **State the problem:** Solve the logarithmic equation $$2 \log_3 x - \log_3 (x - 2) = 2$$ for $x$. 2. **Recall logarithm properties:**

1. **State the problem:** We need to find the ratio of purple to white to blue marbles given 12 purple, 8 white, and 16 blue marbles. 2. **Write the ratio:** The ratio is $12:8:16$

1. **State the problem:** We are given the ratio of students who preferred athletics, rugby, and baseball as 15 : 6 : 4. 2. **Find the total parts in the ratio:** Add the parts of

1. **State the problem:** Solve the equation $x^2 - 2y^2 = 1$ for $y$ in terms of $x$. 2. **Rewrite the equation:** The equation is $x^2 - 2y^2 = 1$.

1. **State the problem:** Solve the equation $3x^2 + 2y^2 = 35$ for one variable or analyze its properties. 2. **Identify the equation type:** This is an equation of an ellipse in

1. **State the problem:** Find the zeros of the function $$f(x) = x^4 - x^3 - 12x^2$$ and sketch its graph. 2. **Factor the function:** Start by factoring out the greatest common f

1. The problem is to find the zeros of the function $$f(x) = x^4 - x^3 - 12x^2$$ and identify the points where the graph crosses the x-axis. 2. The function can be factored as $$f(

1. The problem states the total cost $c$ in pence for $n$ units of electricity is given by the formula: $$c = 17n + 12$$

1. **Stating the problem:** Solve the system of inequalities: $$x^2 + 3x - 4 < 0$$

1. **State the problem:** We are given the equation $$2(3x + 2) + 2y = 66$$ and the relation $$y = 3x + 1$$.

1. The problem appears to be a sum or combined expression involving the numbers and symbols arranged in two columns separated by a vertical bar, with a horizontal double line and t

1. **Stating the problem:** Solve the system of inequalities: $$\begin{cases} x^2 + 3x - 4 < 0 \\ 1 - 3x > 0 \end{cases}$$

1. **State the problem:** Find the domain and range of the function $$y=2x+3$$. 2. **Recall definitions:**

1. The problem asks for the position of point B on the number line between 0 and 1. 2. The number line shows B approximately at the middle between 0 and 1.

1. **State the problem:** Simplify the expression $$\left(\frac{4}{5}\right)^2 - \frac{1}{4}$$. 2. **Apply the exponent:** Square the fraction $$\frac{4}{5}$$ using the rule $$\lef

1. **State the problem:** Calculate the value of $$\left(\frac{4}{5}\right)^2 - \left(\frac{1}{2}\right)^2$$. 2. **Recall the formula:** To square a fraction, square the numerator

1. **State the problem:** We need to find expressions for A, B, and C in the algebraic sums given. 2. **Recall the example:**