🧮 algebra

1. O problema pede para resolver a equação dada (não especificada na mensagem, mas assumiremos que seja uma equação algébrica comum). 2. Para resolver uma equação, usamos a regra f

1. Vamos resolver o primeiro problema: encontrar a área do triângulo OAP formado pelos pontos O(0,0), A(x,0) e P(x,f(x)) com $f(x) = 3 - 4x^2$ e $0 < x < \frac{\sqrt{3}}{2}$. 2. A

1. **Planteamiento del problema:** Simplificar la expresión $$\frac{x}{x^3 - x^2 - 9x + 9}$$. 2. **Factorización del denominador:** Para simplificar, primero factorizamos el polino

1. **State the problem:** We are given that $a - b = b - c = 2$ and need to find the value of $$\frac{(a - b)^2 + (b - c)^2}{(a - c)^2}.$$\n\n2. **Use the given equalities:** Since

1. **State the problem:** Simplify the expression $$\frac{3\sqrt{6} - \sqrt{15}}{\sqrt{6} + 2\sqrt{3}}$$. 2. **Formula and rule:** To simplify a fraction with surds in the denomina

1. **State the problem:** Solve the quadratic equation $2x^2 + 5x + 3 = 0$. 2. **Formula used:** The quadratic formula is given by

1. **Enunciado do problema:** Resolver a equação $$2 \times h(x) + 1 = h(-x)$$ onde $$h(x) = e^x$$ para $$x \in \mathbb{R}$$. 2. **Substituindo a função na equação:**

1. **Problem:** Expand and simplify the expression $(8r - 3)(3r + 8)$. **Formula:** Use distributive property (FOIL): $(a+b)(c+d) = ac + ad + bc + bd$.

1. **State the problem:** Solve the quadratic equation $$x^2 - 4x - 3 = 0$$ for $x$. 2. **Recall the quadratic formula:** For any quadratic equation $$ax^2 + bx + c = 0$$, the solu

1. Solve $3 \cdot (5x - 8)$: Multiply 3 by each term inside the parentheses:

1. **State the problem:** Simplify the expression $3 \cdot (5x - 8)$. 2. **Formula and rules:** Use the distributive property: $a(b + c) = ab + ac$. This means multiply each term i

1. Stating the problem: Solve the equation or system given (not specified, so assuming a generic solve request). 2. Since no specific equation was provided, I cannot solve it witho

1. **Stating the problem:** Simplify the expression $$\sqrt{2} \cdot \sqrt[3]{3}$$. 2. **Recall the rules:**

1. **State the problem:** Find the two values of $x$ such that $$x^2 - 14 = 50$$. 2. **Add 14 to both sides to isolate $x^2$:**

1. **State the problem:** Expand the expression $12x(7x + y)$. 2. **Recall the distributive property:** To expand an expression like $a(b + c)$, multiply $a$ by each term inside th

1. **State the problem:** Solve for $f$ in the equation $$3 = 2f + 15$$. 2. **Isolate the variable term:** Subtract 15 from both sides to get the term with $f$ alone on one side.

1. **State the problem:** Solve the equation $$\frac{48}{c-1} = 3$$ for $c$. 2. **Use the formula:** To solve for $c$, multiply both sides of the equation by the denominator $(c-1)

1. **State the problem:** Expand the expression $3(3a + 2b - c)$. 2. **Formula used:** Use the distributive property of multiplication over addition/subtraction: $$a(b + c + d) = a

1. **State the problem:** Solve the equation $$6 = 18 + \frac{h}{3}$$ for the variable $h$. 2. **Isolate the term with $h$:** To solve for $h$, first subtract 18 from both sides:

1. **State the problem:** Solve the equation $$\frac{x + 51}{7} + 24 = 35$$ for $x$. 2. **Isolate the fraction:** Subtract 24 from both sides to get $$\frac{x + 51}{7} = 35 - 24$$

1. **State the problem:** Factorise the expression $$x^2 - 6x + 2x^2 + 7$$. 2. **Combine like terms:** First, combine the terms with $$x^2$$.