1. Equations Involving Absolute Value

Absolute Value Properties:

• $|p| = |-p|$ for any real number $p$.
• $|p| \geq 0$ for any real number $p$.
• For $|x| = p$ where $p \geq 0$, the solutions are $x = p$ or $x = -p$.
• For $|x| = |p|$ where $p \geq 0$, the solutions are $x = p$ or $x = -p$.

2. For real numbers a, b, and c, any equation that can be simplified to the form $ax^2 + bx + c = 0$, where $( a \neq 0 )$, is known as a quadratic equation.

3. Expressing an equation as a product of its simplest factors is known as factorization.

4. For real numbers a, b, and c, to solve the quadratic equation $ax^2 + bx + c = 0$, where $a ≠ 0 $, you can use the following methods: factorization, completing the square, or the quadratic formula.

5. If the solutions of $ax^2 + bx + c = 0$ using quadratic formula are $x_{1}= \frac{-b + \sqrt{b^{2}-4ac}}{2a}$ and $x_{2}= \frac{-b – \sqrt{b^{2}-4ac}}{2a}$.

i)The sum of the solutions (roots) $x_{1} + x_{2} = \frac{-b}{a}$.

ii) The product of the solutions (roots) $x_{1} \ast x_{2} = \frac{c}{a}$.

6. If the discriminant of a quadratic equation is given by $D = b^2 – 4ac$.

i) If $D > 0$, the quadratic equation has two unique roots.

ii) If $D = 0$, the quadratic equation has exactly one real root.

iii) If ( D < 0 ), the quadratic equation has no real roots.

7. For $( a > 0 )$, $( a^x = a^y )$ if and only if $( x = y )$.

8. Let $𝑥^𝑛 = 𝑘$. If n is even, $x=\pm \sqrt[n]{k}$. If n is odd, $x = \sqrt[n]{k}$.