🧮 algebra

1. The problem asks to find which quadratic equation has a discriminant equal to 0. 2. Recall the discriminant formula for a quadratic equation $ax^2 + bx + c = 0$ is:

1. **State the problem:** Simplify the expression $$a^{\log_{a^2} b^4 + \log_a (7b^2)}$$ where $$a, b > 0$$ and $$a \neq 1$$. 2. **Recall logarithm change of base and properties:**

1. დავწეროთ ფუნქცია: $z=\sqrt{4 - x^2 - y^2}$. 2. ფუნქციის განსაზღვრის არე მოიცავს ყველა იმ წერტილს $(x,y)$, სადაც ფესვის შიგთავსი არ არის უარყოფითი, ანუ:

1. Observing the graph, determine if the following statements are true or false. Justify. 1. The function has no y-intercept.

1. **State the problem:** Solve the quadratic equation $$x^2 + 3x - 9 = 0$$. 2. **Formula used:** The quadratic formula to solve $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b

1. **Problem statement:** Find the locus of a point $P(\alpha, \beta)$ such that the line $y = \alpha x + \beta$ is tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

1. **State the problem:** We are given a system of linear equations: $$2x + 3y = 1$$

1. **State the problem:** We are given the quadratic equation $$x^2 - 5x + 4 = 0$$ and its two real solutions $$x_1 < x_2$$. We need to compute the value of $$x_1 + 2x_2$$. 2. **Re

1. **State the problem:** Solve the linear equation $-5x + y = -7$ for $y$ in terms of $x$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation by a

1. **State the problem:** Solve the linear equation for $y$ given the equation $-5x + y = -7$. 2. **Formula and rules:** To solve for $y$, isolate $y$ on one side of the equation b

1. The problem is to solve the inequality $$\frac{2}{x} \leq -1$$. 2. To solve this, we first note that $$x \neq 0$$ because division by zero is undefined.

1. **Problem statement:** Given points $A(2, -3)$ and $B(-4, 1)$, (a) calculate the length $AB$, and (b) find the equation of the line passing through $A$ and $B$. 2. **Length of s

1. **State the problem:** (a) Given that $$4^3 \times 16^{\frac{1}{4}} = 2^x$$, find the value of $$x$$.

1. **State the problem:** Solve the equation $\sqrt{4n} + 12 = n$ for $n$. 2. **Rewrite the square root:** Note that $\sqrt{4n} = \sqrt{4} \cdot \sqrt{n} = 2\sqrt{n}$.

1. Das Problem: Wir wollen herausfinden, welche gebrochene rationale Funktion zu dem Bild passt, das eine Hyperbel zeigt mit einem senkrechten Asymptoten bei $x=2$ und einem waager

1. **State the problem:** Solve the linear equation $$4 - 3(x + 1) = 2x - 4$$. 2. **Use the distributive property:** Expand terms with parentheses.

1. **State the problem:** Solve the equation $2x - 6 = 4x + 2$ for $x$. 2. **Formula and rules:** To solve linear equations, we isolate the variable on one side by performing the s

1. The problem is to evaluate the expression $\frac{(26.7 - 13 \frac{1}{5})}{1.8} + 0.125$. 2. First, convert the mixed number $13 \frac{1}{5}$ to an improper fraction or decimal:

1. El problema es factorizar el polinomio $$8x^3 - x - 5$$. 2. Para factorizar un polinomio cúbico, podemos intentar encontrar una raíz racional usando el teorema del residuo y lue

1. El problema es factorizar el polinomio $$x^2 + x + 1$$. 2. Para factorizar un trinomio cuadrático de la forma $$ax^2 + bx + c$$, buscamos dos números que multiplicados den $$ac$

1. **State the problem:** We are given the quadratic expression $x^2 + x + 1$ and want to analyze or simplify it. 2. **Formula and rules:** A quadratic expression is generally of t