🧮 algebra

1. **Stating the problem:** Factor the expression $x^2$. 2. **Understanding the expression:** The expression $x^2$ means $x$ multiplied by itself, i.e., $x \times x$.

1. The problem is to find the largest prime factor of 999936. 2. Start by factorizing 999936. Notice that it is even, so divide by 2:

1. State the problem: We want to find the value of $\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49} + 8\sqrt{3}\right)^2$. 2. Simplify each square root:

1. The problem asks us to factorise the cubic polynomial $$x^3 + 8x^2 + 17x + 10$$ into linear factors. 2. To factorise, first try to find rational roots using the Rational Root Th

1. State the problem: We need to find which fraction among $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{6}{11}$, and $\frac{13}{21}$ has the smallest value. 2. To compare fr

1. The problem asks us to evaluate the sum of fractions: $$\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110}$$ 2. To add these

1. The problem asks us to find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8 \sqrt{3}} \right)^2$$. 2. Start by simplifying the terms inside the parentheses:

1. **State the problem:** Factor the quadratic expression $x^2 + 5x + 6$. 2. **Identify coefficients:** This is a quadratic in standard form $ax^2 + bx + c$, where $a=1$, $b=5$, an

1. Stating the problem: Prove that if $$ (1 + x)^n \geq 1 + nx $$ for $$ x > -1 $$ and $$ n \in \mathbb{N} $$, then $$ (1 + x)^{n+1} \geq 1 + (n+1)x $$. 2. Assume the induction hyp

1. Stating the problem: Solve the system of linear equations: $$-5x + 9y - 7y = 0$$

1. **State the problem:** Find the value of $$\left(4\sqrt{4} + 2\sqrt{3} - \sqrt{49 + 8\sqrt{3}}\right)^2.$$\n\n2. **Simplify the terms inside the parentheses:**\n- Calculate $$4\

1. **State the problem:** We need to find which fraction among $\frac{5}{9}$, $\frac{4}{7}$, $\frac{3}{5}$, $\frac{6}{11}$, and $\frac{13}{21}$ has the smallest value. 2. **Method:

1. We start with the problem: Evaluate \( \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} \). 2. First, find a common denomi

1. Problem: For each function, find $f(-x)$ and $-f(-x)$, then compare with $f(x)$ to determine if the function is even, odd, or neither. 2. Function a) $f(x) = x^2 - 4$

1. The **product rule** states that $\log_b(xy) = \log_b x + \log_b y$. This means the log of a product is the sum of the logs. 2. The **quotient rule** states that $\log_b\left(\f

1. Stating the problem: We want to factor the expression $$x^4 + x^2$$. 2. Identify the common factor: Both terms have a factor of $$x^2$$.

1. We are asked to find the domain of each function given. The domain is the set of all real numbers $x$ for which the function is defined. 2. For (a) $f(x) = \sqrt{x+2}$, the expr

1. We are asked to factor each of the given expressions. 2. (a) Factor $4x^2 - 25$.

1. For the pair $f(x) = \frac{1}{x}$ and $g(x) = x$: - Common characteristic: Both are functions that are defined for all $x \neq 0$ (domain excludes zero for $f(x)$), and both are

1. **State the problem:** We need to find the values of $k$ for the quadratic function $$y = kx^2 + (k+3)x - 1$$ such that:\na) The graph cuts the x-axis twice (two distinct real r

1. We are given the half-life of mercury-210 as 600 seconds and an initial mass of 2500 grams. 2. We want to find the remaining mass after 17 minutes.