🧮 algebra

1. The problem is to solve the quadratic equation $x^2 - x - 1 = 0$ for $x$. 2. We use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = 1$, $b = -1$, an

1. **Rationalise the following:** a. $\frac{1 + \sqrt{3}}{2 + \sqrt{3}}$

1. The problem asks to find the result of $50_60$ which is normally interpreted as 50 minus 60. 2. Subtracting 60 from 50: $50 - 60 = -10$.

1. Stated problem: Simplify the expression $$\frac{7ab^3}{10x^2 y^3} : \frac{21a^2 b^2}{15 x^3 y^2}$$. 2. Recall that dividing by a fraction is the same as multiplying by its recip

1. The problem is to solve and explain the expression 50:60. 2. The colon ":" typically represents a ratio in mathematics.

1. **State the problem:** Find the values of $x$ and $y$ such that $$(3x - y) + (x - 3y)i = 5 - i.$$\n\n2. **Identify real and imaginary parts:** For two complex numbers to be equa

1. The problem gives us two exchange rates for pounds (£) to Australian dollars ($) from two travel agents, Aussie Holidays and Kangaroo Travel, both starting at the origin (0,0).

1. Problem: Find the magnitude and slope of vectors $\mathbf{A}$ and $\mathbf{B}$ where $\mathbf{A}=6+ j8$ and $\mathbf{B}=3- j4$. 2. Magnitude of a vector $\mathbf{v}=x+jy$ is giv

1. **State the problem:** We are given two equations: $$a x_1^2 + b x_1 y_1 + c y_1^2 + d = 0$$

1. **State the problem:** Simplify the expression $$\sqrt{x} - 20\sqrt{x^{10}} + 4\sqrt{x^2} - 6\sqrt{x^3} + 8\sqrt{x^4}$$ in radical form.\n\n2. **Rewrite each term using exponent

1. Stating the problem: We need to find the Least Common Multiple (LCM) and Highest Common Factor (HCF, also called GCD) of given numbers. 2. Explanation: The HCF of two or more nu

1. We start with the expression to simplify: $$\sqrt{x} - \sqrt[30]{x^{10}} + \sqrt[12]{x^4} - \sqrt[6]{x^2} + \sqrt[9]{x^3}$$ 2. Rewrite each radical in exponential form using the

1. **State the problem:** Simplify the expression $$\sqrt[3]{54 a^3 b} + \sqrt[3]{7 a^3 b} - a \sqrt[3]{7 b}$$. 2. **Rewrite the cube roots to separate factors with perfect cubes:*

1. **State the problem:** Simplify the expression $$\sqrt[5]{1215a^{5}b} + \sqrt[5]{3a^{5}b} - a\sqrt[5]{3b}$$. 2. **Rewrite each term using properties of radicals:** Because these

1. State the problem: Simplify the expression $$\sqrt{63}vg + \sqrt{112}vg + \sqrt{7}vg$$. 2. Simplify each square root by factoring out perfect squares:

1. **Stating the problem:** We have two semicircle diagrams with numbers placed inside and outside. Our task is to analyze or find a relationship using these numbers. 2. **Observin

1. The problem is to understand the variable $n$ as requested. 2. Without additional context, $n$ is typically used to represent an integer or a count in mathematics.

1. **State the problem:** Given the equation $$(1a + 2i + 152) \left( \frac{-11 + 21}{5} \right) = (-2 - 4i) \left( \frac{(-6 + 2i)^n}{5} - 1 \right)$$

1. **State the problem:** We need to simplify the expression $$(1 + 2i^2)(1 + 2i^5 + 5i^6)$$ and write it in the form $x + yi$ where $x$ and $y$ are real numbers. 2. **Recall power

1. The problem is to find the cube root of 28, which is written as $\sqrt[3]{28}$. 2. Cube root means finding a number $x$ such that $x^3 = 28$.

1. Problem Statement: Given the function $$f(x) = -x^2 + 6x - 11$$, find the following: