Algebra

Expand: $7(2x - 3)$

Factorize the following trinomial: $x^2 - 2x - 8$

Factorize: $100 - x^2$

Factorize the quadratic: $x^2 + 13x + 42$

Factorize by grouping and difference of squares: $p^2 - 4pq + 4q^2 - r^2$.

Factorize the quadratic trinomial: $5x^2 - 13x - 6$.

Expand the product: $(x-3)(x+3)(x^2+9)$.

Expand and simplify the expression $(x+y)^4 - (x-y)^4$.

Factorize $x^2 - y^2 - z^2 + 2yz$.

Factorize the polynomial $x^3 - 3x^2 + 3x - 1$.

What is the discriminant of the quadratic equation $x^2 + 2x + 5 = 0$?

Solve $x^2 + x - 6 = 0$.

If $x^2 = k$ and $k < 0$, how many real solutions are there for $x$?

Determine the $y$-intercept of the parabola $y = x^2 - 5x + 6$.

A rectangular sheet of metal has a length that is $6$ cm longer than its width. A square with side length $2$ cm is cut from each corner, and the sides are folded up to form an open box. Given that the volume of the box is $80$ cm$^3$, find the width of the original sheet of metal.

Express $y = 9 - 4x - x^2$ in the completed square form $a - (x+b)^2$.

If the roots of $x^2 - px + q = 0$ are $\alpha$ and $\beta$, express $\alpha^2\beta + \alpha\beta^2$ in terms of $p$ and $q$.

Solve the equation $x^{-2} - 5x^{-1} + 6 = 0$ for $x$.

If $\sin \theta$ and $\cos \theta$ are the roots of the equation $ax^2 + bx + c = 0$, which of the following identities must hold?

For what range of $m$ does the line $y = mx - 4$ not intersect the parabola $y = x^2$?

Solve for $x$: $11x + 2 = 24$.

Solve for $x$: $7 - 2x = 13$.

Solve for $x$: $\frac{x}{5} - 1 = 2$.

Solve the equation: $3(x + 4) = 2(x - 1)$.

Solve the system: $3x - 4y = 1$ and $2x + 3y = 12$.

The sum of a number $x$ and twice another number $y$ is $18$. If their difference $x - y$ is $3$, find $x$.

Solve: $10 - 2(3x - 1) = 4x$.

Consider the system $x + y + z = 6$, $x + 2y + 3z = 14$, $x + 4y + 7z = 30$. Which of the following is true?

Solve for $x, y, z$: $3x + 2y + z = 10$ $x - y + 2z = 5$ $2x + 3y - z = 1$\nFind $x \cdot y \cdot z$.

If $x, y, z$ satisfy the system $x+2y=5, y+2z=8, z+2x=5$, find $x+y+z$.

Simplify $\sqrt{20}$.

Evaluate $49^{1/2}$.

Solve for $n$: $3^n = 1/9$.

Simplify $(a^2 b^{-3})^2$.

Solve the equation $\sqrt{x+7} = x+1$.

Rationalize the denominator: $\frac{1}{\sqrt{11} - \sqrt{10}}$.

Simplify $\frac{x^{1/2}}{x^{-1/2}}$.

Simplify $\sqrt{x \sqrt{x \sqrt{x^2}}}$.

Solve $2^{2x} - 20(2^x) + 64 = 0$.

If $a = \sqrt{3}+1$, find the value of $a^2 - 2a + 5$.

For $f(x) = 4x - 7$ and $g(x) = \frac{x+7}{4}$, find $fg(x)$.

Which condition ensures that a function $f(x)$ has an inverse?

If $f(x) = 3x - k$ and $f^{-1}(11) = 5$, find the value of the constant $k$.

Functions $f$ and $g$ are such that $f(x) = \sin(x)$ and $g(x) = 2x$. Find the value of $gf(\pi/6)$.

The diagram shows the graph of the function $f(x) = x^2 - 4$ for $x \ge 0$. The function $g$ is defined by $g(x) = 2x + 1$ for $x \in \mathbb{R}$. Find the value of $x$ for which $f(g(x)) = 21$.

Determine the composite function $gh(x)$ if $g(x) = x^2 + 1$ and $h(x) = \sqrt{x-1}$ for $x \geq 1$.

If $f(x) = 5x + 2$, solve the equation $f(x) = f^{-1}(x)$.

A mapping is defined by $f: x \mapsto 4x - x^2$ for $x \in \mathbb{R}, x \ge 2$. Find the domain and range of $f^{-1}$.

The temperature of a cooling object is $H(t) = 20 + 60e^{-0.1t}$. To find the time $t$ for a given temperature $H$, we use the inverse function. Find $H^{-1}(50)$.

Let $f(x) = \frac{2x-1}{x+4}$ for $x \neq -4$. If $f(g(x)) = x$, find the expression for $g(x)$.

Simplify $e^{-\ln x}$ for $x > 0$.

What is the value of $\log_5 1$?

Identify the $y$-intercept of the graph $y = 5 e^{2x}$.

Solve for $x$: $3^{x+1} = 27$.

What is the result of $\log_2 (\log_3 81)$?

Given $\ln y = 2x + 3$, express $y$ in terms of $x$.

Solve the inequality $0.8^n < 0.1$. Find the smallest integer $n$.

A radioactive isotope decays according to $M = M_0 (0.5)^{t/h}$, where $h$ is the half-life. If the mass reduces to 20% of its original mass in 15 years, find the half-life $h$ correct to 1 decimal place.

The number of people $N$ who have heard a rumor after $t$ days is given by $N = \frac{500}{1 + 49e^{-0.2t}}$. Determine the number of days it takes for 250 people to have heard the rumor.

Solve the equation $\ln(x+2) + \ln(x-1) = \ln(10)$ for $x$.

Find the common difference in the sequence $10, 6, 2, -2, \dots$.

Calculate the sum to infinity for $a = 6$ and $r = -1/2$.

What is the sum of the first 4 terms of the sequence $u_n = 2n + 1$?

In a geometric sequence, $a = 4$ and $u_2 = 12$. Find $r$.

The sum of the first $n$ terms of a sequence is given by $S_n = 3n^2 - n$. Find the 5th term of the sequence.

Evaluate the summation of the first five cubic numbers: $\sum_{n=1}^{5} n^3$.

Calculate the sum to infinity of the series $1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \dots$.

The 1st, 2nd, and 5th terms of an arithmetic progression are the first three terms of a geometric progression. If the first term of the AP is 4, find the common difference $d$ (where $d \neq 0$).

Given $S_n = \frac{n}{2}(3n+1)$ is the sum of an arithmetic progression, find the common difference $d$.

In an infinite geometric series where the first term is $k$ and the common ratio is $k$, the sum to infinity is 1. Find $k$.