🧮 algebra

1. **State the problem:** We need to resolve the rational expression $$\frac{7x + 2}{(2x - 3)(x + 1)^3}$$ into partial fractions.

1. **State the problem:** Solve the quadratic equation $$3x^2 - x - 10 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equ

1. **State the problem:** Simplify or factor the quadratic expression $3x^2 - x - 10$. 2. **Formula and rules:** To factor a quadratic $ax^2 + bx + c$, find two numbers that multip

1. The problem asks to resolve given rational expressions into partial fractions. 2. Partial fraction decomposition involves expressing a rational function as a sum of simpler frac

1. **State the problem:** Simplify the expression $$\frac{173}{11} - \frac{93}{4} + \frac{4}{5}$$. 2. **Find a common denominator:** The denominators are 11, 4, and 5. The least co

1. **Resolve into partial fractions:** (i) $$\frac{6x - 10}{x^2 - 2x - 3}$$

1. **Problem (a): Simplify** $4p^3q^5 \times 6p^2q$ - We multiply the coefficients: $4 \times 6 = 24$

1. **State the problem:** We need to find the value of $x$ given the equation $\log_x 125 = 3$. 2. **Recall the definition of logarithm:** The equation $\log_a b = c$ means that $a

1. **State the problem:** Solve the equation $x^2 + 25 = 0$ for $x$. 2. **Recall the formula:** To solve quadratic equations of the form $x^2 + c = 0$, isolate $x^2$ and then take

1. **State the problem:** We are given the system of inequalities: $$\begin{cases} x - y \geq 4 \\ x + y \leq 6 \end{cases}$$

1. **State the problem:** We are asked to analyze the function $x^2$. 2. **Formula and rules:** The function is $y = x^2$, which is a quadratic function. Quadratic functions have t

1. **State the problem:** Solve the quadratic equation $5x^2 - 35x = 0$. 2. **Formula and rules:** To solve quadratic equations, we can factorize or use the quadratic formula. Here

1. **State the problem:** Add the mixed fractions $3 \frac{4}{9}$ and $2 \frac{2}{4}$.\n\n2. **Convert mixed fractions to improper fractions:**\n- $3 \frac{4}{9} = \frac{3 \times 9

1. The problem asks to find $fog(-1)$ given that $g(3) = 9$ and that $f$ and $g$ are functions involved in a composition. 2. Function composition $fog(x)$ means $f(g(x))$, so we fi

1. **State the problem:** Solve the quadratic equation $7x^2 - 28 = 0$ for $x$. 2. **Formula and rules:** To solve quadratic equations of the form $ax^2 + bx + c = 0$, we can isola

1. **State the problem:** Solve the quadratic equation $$3x^2 + 3x - 60 = 0$$. 2. **Formula and rules:** The general quadratic equation is $$ax^2 + bx + c = 0$$. The solutions can

1. **State the problem:** Solve the quadratic equation $$3x^2 + x - 60 = 0$$. 2. **Formula used:** To solve a quadratic equation of the form $$ax^2 + bx + c = 0$$, use the quadrati

1. The problem is to calculate $\frac{3}{4} + \frac{2}{5}$ and express the answer as a mixed number. 2. To add fractions, we need a common denominator. The denominators are 4 and 5

1. The problem is to evaluate the expression $32^{\frac{1}{2}}$. 2. Recall that raising a number to the power of $\frac{1}{2}$ is equivalent to taking the square root of that numbe

1. The problem is to simplify the fraction $\frac{168}{14}$. 2. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).

1. **State the problem:** Calculate the sum of one seventh and two fifths, i.e., $\frac{1}{7} + \frac{2}{5}$.\n\n2. **Formula and rules:** To add fractions, they must have a common