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

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

Two non-adjacent angles formed by intersecting lines; they are always congruent

Two angles whose measures add up to 180°

Two angles whose measures add up to 90°

180°

360°

(n - 2) × 180°

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

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

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

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