🧮 algebra

1. **State the problem:** Rationalise the denominator of the expression $$\frac{4}{\sqrt{7} - \sqrt{11}}.$$\n\n2. **Recall the formula:** To rationalise a denominator with a differ

1. Problem: Find the product of the roots of the equation $$(\log_3 x)^2 - 3 \log_3 x - 4 = 0.$$\n\n2. Step 1: Let $y = \log_3 x$. Then the equation becomes $$y^2 - 3y - 4 = 0.$$\n

1. **Evaluate** $|3^2 - 5^2|$. First, calculate the squares:

1. **State the problem:** Simplify the expression $$\frac{a^2bc^3}{ab^2c^{-1}}$$. 2. **Recall the rules:** When dividing powers with the same base, subtract the exponents: $$\frac{

1. Problem: Çözmemiz gereken denklem $\log_{k+2}(x^2 + 3x - 6) = 1$. 2. Logaritmanın tanımına göre, $\log_a b = c$ ise $a^c = b$ olur.

1. Stating the problem: Simplify the expression $$7 \times \frac{1}{3} + 5 \times \frac{4}{9} - 4 \times \frac{4}{9}$$. 2. Use the distributive property and multiply each integer b

1. **State the problem:** Simplify the expression $\frac{\ln(3x)}{3x}$. 2. **Recall the properties:** The expression is a fraction with a logarithm in the numerator and a product i

1. The problem asks: When using the quadratic formula, if the coefficient $b$ is negative, does the $-b$ in the formula become positive $b$ when substituted? 2. Recall the quadrati

1. **State the problem:** Solve the quadratic equation $$4x^2 + 16x + 7 = 0$$ using the quadratic formula. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$

1. **State the problem:** We have the function $$f(x) = x^4 + \frac{1}{3}x^3 - 8x^2 + ax + \frac{17}{3}$$ where $$a$$ is a constant.

1. **Problem:** Find the intersection of sets $A = \{x \mid x \leq 0 \text{ or } x > 1\}$ and $B = \{-2,0,1,2\}$.\n\n2. **Formula and rules:** The intersection $A \cap B$ contains

1. The problem is to analyze the function given in the document link, but since the link is inaccessible here, I will demonstrate solving a typical algebra problem involving a quad

1. **State the problem:** Given functions $g(x) = \frac{1-x}{x}$ and the composition $g \circ f(x) = \frac{1}{x+1}$, find the inverse of the composition, i.e., $(g \circ f)^{-1}$.

1. **State the problem:** Solve the system of equations using Cramer's rule. Suppose the system is:

1. **State the problem:** Simplify the expression $$\left[(1 - a)^2 + (2 - b)^2\right] + (2a - b)(2a + b) + (1 + a)^2 + 4b - (a - b)(a + b)$$. 2. **Recall formulas and rules:**

1. **Stating the problem:** We are given a quadratic equation and need to find its roots.

1. **State the problem:** We have an arithmetic sequence with terms $(2k+2), (5k+3), \ldots, 518$ and the sum of all terms is 5628. We need to find the value of $k$. 2. **Identify

1. **State the problem:** We need to find the gradient (slope) and the y-intercept of the line shown on the graph. 2. **Identify points on the line:** From the graph description, t

1. **Problem statement:** (a) Show that for small $x$ in radians, $1 - \cos^2(2x) \approx 4x^2 - 4x^4$.

1. **State the problem:** We have an arithmetic sequence with terms $2k + 2$, $5k + 3$, ..., $518$ and the sum of all terms is $5628$. We need to find the value of $k$.

1. **State the problem:** We need to find the gradient (slope) and the y-intercept of the line shown on the graph. 2. **Identify points on the line:** From the description, the lin