🧮 algebra

1. **State the problem:** Simplify and solve the equation $$\frac{3}{x+5} - \frac{1}{x^2 + 2x - 15} = \frac{1}{x+5}$$. 2. **Factor the quadratic denominator:** Note that $$x^2 + 2x

1. The problem is to find the sum of integers from 1 to $n$. 2. The formula to find the sum of the first $n$ integers is given by the arithmetic series sum formula:

1. **State the problem:** Evaluate the expression $$[(-3)(4) + 5(-3)][4(-5) - 5(-3)]^2$$. 2. **Apply multiplication inside the brackets:**

1. The problem is to graph the function $y = x^2 - 2x + 1$ and understand its shape. 2. The function is a quadratic polynomial, which graphs as a parabola. The general form is $y =

1. **Problem statement:** Simplify the expression $$9x - (3 - 3x)$$ by first removing the parentheses and then combining like terms. 2. **Formula and rules:** When removing parenth

1. **Problem Statement:** Find the value of $n$ for the expression $$\frac{(n+2)!}{(n-1)!}.$$ 2. **Recall the factorial definition:**

1. **Problem Statement:** Evaluate $$\frac{(n+2)!}{(n-1)!}$$ for a general $n$. 2. **Recall the factorial definition:**

1. **State the problem:** Evaluate the expression $$\frac{x^2 - 4x + 2}{x}$$ at $$x = 4$$. 2. **Write the expression:** $$\frac{x^2 - 4x + 2}{x}$$.

1. **State the problem:** Simplify the expression $$(2x^2 + yx^2)(3x^2 + 3x^2).$$ 2. **Combine like terms inside the parentheses:**

1. **Problem Statement:** Find the value of $n$ if $$\frac{n!}{(n-3)!} = 990.$$

1. **State the problem:** Factorize the polynomial $x^4 - 13x^2 - 36$. 2. **Identify the type of polynomial:** This is a quartic polynomial in terms of $x$, but notice it can be tr

1. **State the problem:** Factor the polynomial $$x^4 + 6x^3 + 16x^2 + 18x + 7$$. 2. **Recall the formula and rules:** To factor a quartic polynomial, try to express it as a produc

1. The problem is to graph the function $y = x^2$, which is a quadratic function. 2. The general form of a quadratic function is $y = ax^2 + bx + c$. Here, $a=1$, $b=0$, and $c=0$.

1. **Problem statement:** Solve the inequality $-7z \geq 14$ in the set of real numbers. 2. **Formula and rules:** To solve inequalities, isolate the variable on one side. Remember

1. The problem is to analyze the function $$S_{xx}(\omega) = \frac{\omega^2 + 9}{\omega^4 + 5\omega^2 + 4}$$ and understand its behavior. 2. The formula given is a rational functio

1. **Problem 42:** Find the value of $k - a$ given that the line $y = kx - 7$ and the parabola $y = ax^2 - 13x + 17$ intersect at points with abscissas 4 and 2. 2. **Step 1:** Sinc

1. The problem is to graph a quadratic function that produces a U-shaped curve with a vertex (lowest point) around $y=6$ and x-values roughly between $-1.5$ and $1.5$. 2. The gener

1. **State the problem:** Simplify the expression $$\frac{f-1}{e g} \div \frac{f g}{g+2} = \frac{g}{g}$$. 2. **Recall the division of fractions rule:** Dividing by a fraction is th

1. **Problem Statement:** Find the prime factorisation of the numbers 72, 756, 187, and 630, expressing each answer in index notation. 2. **Formula and Rules:** Prime factorisation

1. **Stating the problem:** We are given a piecewise function $$R(\tau)$$ defined as: $$R(\tau) = \begin{cases} \lambda^2, & |\tau| > \lambda^2 \\ \lambda(1 - 3|\tau|), & |\tau| \l

1) Solve $\frac{x - 7}{x - 1} < 0$. Step 1: Identify critical points where numerator or denominator is zero: $x=7$ and $x=1$.