Algebra & Geometry Reference Sheet

ax² + bx + c = 0, where a ≠ 0

x = (-b ± √(b² - 4ac)) / 2a

Often memorized as a song

b² - 4ac (the expression under the radical in the quadratic formula)

Two distinct real solutions

Exactly one real solution (a repeated root)

Two complex (imaginary) solutions with no real solutions

y = a(x - h)² + k, where (h, k) is the vertex

x = -b / (2a)

Rewriting a quadratic in vertex form by creating a perfect square trinomial

A parabola

The highest exponent of the variable in the polynomial

Every non-constant polynomial with complex coefficients has at least one complex root

Exactly n roots (counting multiplicity, in the complex numbers)

Dividing a polynomial by a linear factor of the form (x - c)

When a polynomial f(x) is divided by (x - c), the remainder equals f(c)

If f(c) = 0, then (x - c) is a factor of f(x)

√(-1)

-1

Circle, ellipse, parabola, and hyperbola

(x - h)² + (y - k)² = r²

The set of all points where the sum of distances to two foci is constant

It looks like a stretched circle

x²/a² + y²/b² = 1

The set of all points where the absolute difference of distances to two foci is constant

A circle

A parabola

0 (zero)

A circle is a perfectly 'round' ellipse

y = a(x - h)² + k, where (h, k) is the vertex

The vertex coordinates appear right inside the parentheses

x²/a² - y²/b² = 1

The minus sign is what separates a hyperbola from an ellipse

y = ±(b/a)x, two lines the hyperbola's branches approach but never touch

It is a fixed line such that every point on the parabola is equidistant from the directrix and the focus

Think of it as the focus's mirror-image partner, a line instead of a point

Euclid

Through a point not on a line, exactly one parallel line can be drawn to the given line

A statement accepted as true without proof

A statement that has been proven true using postulates, definitions, and logical reasoning

Non-adjacent angles from intersecting lines (always congruent)

Two angles whose measures add up to 180°

Two angles whose measures add up to 90°

180°

360°

(n - 2) × 180°

Corresponding angles are congruent

They sit in the same relative position at each intersection

Alternate interior angles are congruent

They lie between the parallel lines, on opposite sides of the transversal

In a right triangle, a² + b² = c², where c is the hypotenuse

SSS, SAS, ASA, AAS, and HL (Hypotenuse-Leg for right triangles)

SSA (donkey theorem) does NOT work

Side-Side-Side: if all three sides of one triangle equal the corresponding sides of another, the triangles are congruent

Side-Angle-Side: two sides and the included angle are equal

AAA only proves similarity (same shape), not congruence (same size), because the triangles could be different sizes

They have the same shape (equal corresponding angles) but not necessarily the same size; corresponding sides are proportional

1 : √3 : 2

1 : 1 : √2

The sum of any two sides of a triangle must be greater than the third side

If two sides of a triangle are congruent, then the angles opposite those sides are also congruent

3-4-5, 5-12-13, and 8-15-17

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles

A straightedge (unmarked ruler) and a compass

Draw arcs of equal radius from both endpoints; connect the two intersection points to find the midpoint

Draw an arc from the vertex, then arcs from the two intersection points; draw a ray through the new intersection

A line that passes through the midpoint of a segment at a 90° angle

Trisecting an angle, doubling the cube, and squaring the circle

All proven impossible in the 19th century

The point where the three perpendicular bisectors of the sides intersect; it is the center of the circumscribed circle

The point where the three angle bisectors intersect; it is the center of the inscribed circle

Draw an arc from the point crossing the line at two spots, then draw arcs from those two spots to meet on the opposite side; connect that meeting point to the original point

Draw an arc across both rays of the original angle, then draw the same-radius arc on a new ray; measure the arc's width and mark it on the new arc to fix the second ray

The compass preserves distances, so it can transfer the arc's 'width' to a new location

Draw a circle centered at each endpoint with radius equal to the segment; connect both endpoints to one of the circles' intersection points

This is Euclid's very first proposition in the Elements

The equilateral triangle, square, regular pentagon, and regular hexagon (and others built from these by repeated bisection)

A regular heptagon (7 sides) is famously NOT among them

It requires constructing a length equal to the cube root of 2, which cannot be built from straight lines and circles alone

Compass-and-straightedge constructions can only produce certain kinds of numbers

d = √((x₂ - x₁)² + (y₂ - y₁)²)

It comes from the Pythagorean Theorem

((x₁ + x₂)/2, (y₁ + y₂)/2)

m = (y₂ - y₁) / (x₂ - x₁), or 'rise over run'

y = mx + b, where m is the slope and b is the y-intercept

y - y₁ = m(x - x₁)

Their slopes are negative reciprocals (product = -1)

They have equal slopes

Ax + By = C, where A, B, and C are integers

The slope is undefined, since the line has no horizontal change

The slope formula would require dividing by a run of zero

The slope is zero, since the line has no vertical change

Set the distance from any point (x, y) to the center (h, k) equal to the radius r, which gives (x - h)² + (y - k)² = r²

Show that both pairs of opposite sides have equal slopes (are parallel), often by using the slope formula on each side

Parallel sides mean matching slopes

A = ½bh (half base times height)

A = πr²

C = 2πr (or πd)

V = lwh (length × width × height)

V = πr²h

V = ⅓πr²h

One-third of a cylinder

V = (4/3)πr³

SA = 4πr²

V = ⅓Bh (one-third base area times height)

Same relationship to prism as cone to cylinder

A = ½(b₁ + b₂)h (half the sum of the bases times the height)

A = bh (base times height, where height is perpendicular to base)

LA = 2πrh

Imagine 'unrolling' the cylinder into a rectangle