🧮 algebra

1. Problema 54: Risolvere l'equazione $$\frac{1}{x^2 - 2x + 1} + \frac{2}{x^2 - 3x + 2} = - \frac{2}{2 - x}$$ 2. Notiamo che $$x^2 - 2x + 1 = (x-1)^2$$ e $$x^2 - 3x + 2 = (x-1)(x-2

1. **State the problem:** Jill and Brian together invested $1900 in two accounts. Jill's account earns 4% interest, Brian's earns 3%, and the total interest earned is $70. We need

1. **Problem Statement:** We are given that $f(1) = f(-5)$ and want to determine if this implies the $h$ value of the vertex of the parabola is $-3$. 2. **Recall the vertex form of

1. **State the problem:** We have a quadratic function $$f(x) = ax^2 + 12x + c$$ with constants $$a$$ and $$c$$. The parabola opens upward, so $$a > 0$$. The vertex is at $$(h,k)$$

1. **State the problem:** Polly’s mother packed party bags with 4 sweets and 3 biscuits each. There are 100 sweets in total. We need to find how many biscuits she packed. 2. **Defi

1. **State the problem:** Polly’s mother packed party bags with 4 sweets and 3 biscuits each. There are 100 sweets in total. We need to find how many biscuits she packed. 2. **Defi

1. **State the problem:** We need to find the equation of a line that is perpendicular to the line $$y = \frac{1}{6}x + 3$$ and passes through the point $$(-3, 23)$$. 2. **Recall t

1. **State the problem:** Find the equation of the line perpendicular to $y = -\frac{2}{3} x$ that passes through the point $(4,-8)$. 2. **Recall the slope of the given line:** The

1. **State the problem:** Simplify the expression $$16^{-\frac{3}{2}}$$. 2. **Recall the rules:** For any positive number $a$ and rational exponent $m/n$, $$a^{\frac{m}{n}} = \sqrt

1. **State the problem:** We need to find the equation of an increasing S-shaped curve (sigmoid-like) with an x-intercept at $(-1, 0)$ and passing through the point $(2, 5)$. 2. **

1. **State the problem:** We are given a quartic function of the form $$y = a(x - b)^2(x - c)^2$$ and a graph with x-intercepts at $(-2,0)$ and $(1,0)$, and a point $(0,2)$ on the

1. **State the problem:** Find the equations of lines passing through the point $(10,-2)$ parallel to the given lines $4x-2y+1=0$ and $x+5y-10=0$. 2. **Find the slope of each given

1. **State the problem:** Find the equation of the line passing through the point $(10,-2)$ that is parallel to a given line. 2. **Identify the slope of the given line:** To find t

1. **Problem Statement:** Find the value of $x$ that satisfies the equation $$\frac{2x+3}{5} = \frac{3x-1}{4}.$$

1. **State the problem:** Simplify the expression $$\frac{x^2-25}{x^2+5x} \div \frac{xy+6x-5y-30}{5x-15}$$. 2. **Rewrite division as multiplication by reciprocal:**

1. **State the problem:** Three girls initially pay $100$ each, totaling $$300$$, for a motel room. The correct charge is $$250$$, so the clerk gives $$50$$ to the attendant to ret

1. **State the problem:** Simplify the expression $$\frac{x^2-25}{x^2+5x} \div \frac{xy+6x-5y-30}{5x-15}$$. 2. **Recall the division of fractions rule:** Dividing by a fraction is

1. **State the problem** Solve the equation $16 - 2t = 5t + 9$.

1. **State the problem:** Solve the system of equations:

1. **State the problem:** Simplify the expression $$\frac{2x^2 - 8}{4x}$$. 2. **Formula and rules:** To simplify a rational expression, factor numerator and denominator and cancel

1. **State the problem:** Simplify the expression $$6 \div 2(1+2)$$. 2. **Apply the order of operations (PEMDAS/BODMAS):**