For any equation \(ax^2 + bx + c = 0\), the solutions are given by the Quadratic Formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The expression \(D = b^2 - 4ac\) is called the discriminant. If \(D > 0\): two real roots; \(D = 0\): one root; \(D < 0\): no real roots.

A complex number is written as \(a + bi\), where \(i = \sqrt{-1}\), so \(i^2 = -1\).
To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. The conjugate of \(a+bi\) is \(a-bi\), and \((a+bi)(a-bi) = a^2 + b^2\).

Remainder Theorem: When polynomial \(p(x)\) is divided by \((x-k)\), the remainder equals \(p(k)\).
Factor Theorem: \((x-k)\) is a factor of \(p(x)\) if and only if \(p(k) = 0\).

\(\log_b x = y\) means \(b^y = x\). Key rules:
\(\log_b(xy) = \log_b x + \log_b y\)   \(\log_b(x^n) = n\log_b x\)
\(\log_b b = 1\)   \(\log_b 1 = 0\)

\(a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\)
For negative bases, odd roots are defined: \(\sqrt[n]{-a} = -\sqrt[n]{a}\) when \(n\) is odd.

To complete the square for \(y = x^2 + bx + c\):
Add and subtract \(\left(\dfrac{b}{2}\right)^2\) to create a perfect square trinomial.
Result: \(y = \left(x + \dfrac{b}{2}\right)^2 + \left(c - \dfrac{b^2}{4}\right)\).
The vertex is at \(\left(-\dfrac{b}{2},\ c - \dfrac{b^2}{4}\right)\).

An infinite geometric series \(a + ar + ar^2 + \cdots\) converges when \(|r| < 1\):
\[ S = \frac{a}{1 - r} \] If \(|r| \geq 1\), the series diverges (no finite sum).

To find \(f^{-1}(x)\): ① replace \(f(x)\) with \(y\), ② swap \(x\) and \(y\), ③ solve for \(y\).
Verify: \(f(f^{-1}(x)) = x\).

For a linear-quadratic system, substitute the linear equation into the quadratic, then solve the resulting quadratic. Each solution gives one intersection point.

Exponential decay: \(A = A_0(1 - r)^t\)
where \(A_0\) = initial amount, \(r\) = decay rate (as a decimal), \(t\) = time.

If \(\triangle ABC \sim \triangle DEF\), then corresponding sides are proportional:
\[\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \quad \text{(scale factor)}\]

Inscribed Angle Theorem: An inscribed angle equals half the central angle subtending the same arc.
\[\angle_{\text{inscribed}} = \frac{1}{2} \times \angle_{\text{central}}\]

Distance between \((x_1, y_1)\) and \((x_2, y_2)\):
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] This is the Pythagorean theorem applied to coordinates.

Cylinder: radius \(r\), height \(h\)
Volume: \(V = \pi r^2 h\)
Total Surface Area: \(A = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)\)

When a transversal cuts two parallel lines:
• Alternate interior angles are equal.
• Corresponding angles are equal.
• Co-interior (same-side) angles are supplementary (sum = 180°).

Midpoint of segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\):
\[M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right)\]

SSS — three pairs of equal sides
SAS — two sides and the included angle
ASA — two angles and the included side
AAS — two angles and a non-included side
HL — hypotenuse and leg (right triangles only)

For angle \(\theta\) in a right triangle — SOH-CAH-TOA:
\[\sin\theta = \frac{\text{opp}}{\text{hyp}}, \quad \cos\theta = \frac{\text{adj}}{\text{hyp}}, \quad \tan\theta = \frac{\text{opp}}{\text{adj}}\]

Sum of interior angles of an \(n\)-gon: \(S = (n-2) \times 180°\)
Each interior angle of a regular \(n\)-gon: \(\dfrac{(n-2) \times 180°}{n}\)
Each exterior angle: \(\dfrac{360°}{n}\)

Circle with center \((h, k)\) and radius \(r\):
\[(x - h)^2 + (y - k)^2 = r^2\] To find center and radius from general form, complete the square in both \(x\) and \(y\).