The Quadratic Formula solves any equation of the form \(ax^2 + bx + c = 0\):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The discriminant \(D = b^2 - 4ac\) tells you the nature of roots:
\(D > 0\): two real roots  |  \(D = 0\): one real root  |  \(D < 0\): two complex roots

A complex number has the form \(a + bi\), where \(i = \sqrt{-1}\), so \(i^2 = -1\).
To multiply, use FOIL and replace \(i^2\) with \(-1\).
To divide, multiply numerator and denominator by the conjugate of the denominator.

Remainder Theorem: When a 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 \(p(k) = 0\).

Key log rules:
\(\log_b(xy) = \log_b x + \log_b y\)  |  \(\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y\)
\(\log_b(x^n) = n\log_b x\)  |  \(\log_b b = 1\)  |  \(\log_b 1 = 0\)

A rational exponent: \(a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\)
Properties: \(a^{-m/n} = \dfrac{1}{a^{m/n}}\),   \(a^{1/n} = \sqrt[n]{a}\)

For a system with a linear and a quadratic equation, substitute the linear equation into the quadratic, then solve. The number of intersection points equals the number of solutions.

Geometric Series Sum: For \(n\) terms with first term \(a\) and common ratio \(r \neq 1\):
\[ S_n = \frac{a(1 - r^n)}{1 - r} \] Infinite Geometric Series (if \(|r| < 1\)): \(S = \dfrac{a}{1 - r}\)

To complete the square for \(ax^2 + bx + c\), rewrite as \(a(x + h)^2 + k\):
1. Factor out \(a\) from the \(x\)-terms
2. Take half the coefficient of \(x\), square it, add and subtract inside
The vertex form \(a(x-h)^2 + k\) gives vertex at \((h, k)\).

To find the inverse \(f^{-1}(x)\) of a function \(f(x)\):
1. Replace \(f(x)\) with \(y\)
2. Swap \(x\) and \(y\)
3. Solve for \(y\) — that's \(f^{-1}(x)\)
Verify: \(f(f^{-1}(x)) = x\) and \(f^{-1}(f(x)) = x\).

Exponential model: \(A = A_0 \cdot b^t\)
Growth: \(b > 1\),   Decay: \(0 < b < 1\).
Common form: \(A = A_0(1 \pm r)^t\) where \(r\) is the rate.

Two triangles are similar (AA, SAS, SSS similarity) if corresponding angles are equal and sides are proportional.
If \(\triangle ABC \sim \triangle DEF\), then \(\dfrac{AB}{DE} = \dfrac{BC}{EF} = \dfrac{AC}{DF}\) (scale factor).

Inscribed Angle Theorem: An inscribed angle is half the central angle that subtends the same arc.
\(\angle\text{inscribed} = \dfrac{1}{2} \times \text{arc}\)
A semicircle subtends a 90° inscribed angle.

In a right triangle with legs \(a, b\) and hypotenuse \(c\):
\[ a^2 + b^2 = c^2 \] Distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

Cylinder:
Volume: \(V = \pi r^2 h\)
Lateral Surface Area: \(A_L = 2\pi r h\)
Total Surface Area: \(A_T = 2\pi r^2 + 2\pi r h = 2\pi r(r + h)\)

When a transversal cuts parallel lines:
• Alternate interior angles: equal   • Co-interior (same-side) angles: supplementary (\(=180°\))
• Corresponding angles: equal   • Alternate exterior angles: equal

The midpoint of segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ M = \left(\frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2}\right) \] If \(M\) is known and one endpoint is known, use algebra to find the other.

Congruence criteria (two triangles are congruent if):
SSS: three pairs of equal sides
SAS: two sides and the included angle equal
ASA: two angles and the included side equal
AAS: two angles and a non-included side equal
HL: hypotenuse and leg equal (right triangles only)

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

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 of a regular \(n\)-gon: \(\dfrac{360°}{n}\)

Equation of a circle with center \((h, k)\) and radius \(r\):
\[ (x - h)^2 + (y - k)^2 = r^2 \] Standard form can be obtained by completing the square in \(x\) and \(y\).