Math Resources Algebra Formulas

Algebraic Formula Sheet Arithmetic Operations ac + bc = c(a + b) a/b * c/d = ac/bd (a/b)/c = a/(bc) a/(b/c) = ac/b a/b + c/d = (ad + bc)/bd a/b - c/d = (ad - bc)/bd a - b / (c - d) = (b - a)/(d - c) a + b / c = a/c + b/c ab + ac / a = b + c, a ≠ 0 a/b / c/d = ad/bc Properties of Exponents xⁿxᵐ = xⁿ⁺ᵐ x⁰ = 1, x ≠ 0 (xⁿ)ᵐ = xⁿᵐ xⁿ/yⁿ = (x/y)ⁿ (xy)ⁿ = xⁿyⁿ 1/x⁻ⁿ = xⁿ xⁿ/xᵐ = xⁿ⁻ᵐ (x/y)⁻ⁿ = (y/x)ⁿ = yⁿ/xⁿ x⁻ⁿ = 1/xⁿ Properties of Radicals ⁿ√x = x¹/ⁿ ⁿ√x/ⁿ√y = ⁿ√(x/y) ⁿ√x ᵐ√y = ⁿᵐ√(xᵐyⁿ) ⁿ√(xⁿ) = |x|, if n is even ⁿ√(xⁿ) = x, if n is odd ⁿ√xᵐ√x = ⁿᵐ√(xⁿ⁺ᵐ) Properties of Inequalities If a < b then a + c < b + c and a - c < b - c If a < b and c > 0 then ac < bc and a/c < b/c If a < b and c < 0 then ac > bc and a/c > b/c Properties of Absolute Value |x| = { x if x ≥ 0; -x if x < 0 } |x| ≥ 0 |-x| = |x| |xy| = |x||y| |x/y| = |x|/|y| |x + y| ≤ |x| + |y| Triangle Inequality |x - y| ≥ ||x| - |y|| Reverse Triangle Inequality Distance Formula Given two points, P₁ = (x₁, y₁) and P₂ = (x₂, y₂), the distance between the two can be found by: d(P₁, P₂) = √((x₂ - x₁)² + (y₂ - y₁)²) Number Classifications Natural Numbers : N = {1, 2, 3, 4, 5, . . .} Whole Numbers : {0, 1, 2, 3, 4, 5, . . .} Integers : Z = {. . . ,-3, -2, -1, 0, 1, 2, 3, . . .} Rationals : Q = {All numbers that can be written as a fraction with an integer numerator and a nonzero integer denominator, a/b} Irrationals : {All numbers that cannot be expressed as the ratio of two integers, for example √3, ⁵√27, and π} Real Numbers : R = {All numbers that are either a rational or an irrational number} Logarithms and Log Properties Definition y = logₐx is equivalent to x = aʸ Example log₂16 = 4 because 2⁴ = 16 Special Logarithms ln x = logₑx natural log where e=2.718281828... log x = log₁₀x common log Logarithm Properties logₐ1 = 0 logₐa = 1 logₐaˣ = x a^(logₐx) = x ln eˣ = x e^(ln x) = x logₐ(xᵏ) = k logₐx logₐ(xy) = logₐx + logₐy logₐ(x/y) = logₐx - logₐy Factoring xa + xb = x(a + b) x² + y³ = (x + y)(x² - xy + y²) x² - y² = (x + y)(x - y) x³ - y³ = (x - y)(x² + xy + y²) x² + 2xy + y² = (x + y)² x²ⁿ - y²ⁿ = (xⁿ - yⁿ)(xⁿ + yⁿ) x² - 2xy + y² = (x - y)² If n is odd then, x³ + 3x²y + 3xy² + y³ = (x + y)³ xⁿ - yⁿ = (x - y)(xⁿ⁻¹ + xⁿ⁻²y + ... + yⁿ⁻¹) x³ - 3x²y + 3xy² - y³ = (x - y)³ xⁿ + yⁿ = (x + y)(xⁿ⁻¹ - xⁿ⁻²y + xⁿ⁻³y² - ... - yⁿ⁻¹) Linear Functions and Formulas Examples of Linear Functions [Image of a line passing through the origin with a positive slope, labeled y = x. The graph is titled "linear function".] [Image of a horizontal line at y=1. The graph is titled "constant function".] Constant Function This graph is a horizontal line passing through the points (x,c) with slope m = 0 : y = c or f(x) = c Slope (a.k.a Rate of Change) The slope m of the line passing through the points (x₁,y₁) and (x₂,y₂) is: m = Δy/Δx = (y₂ - y₁)/(x₂ - x₁) = rise/run Linear Function/Slope-intercept form This graph is a line with slope m and y-intercept(0,b) : y = mx + b or f(x) = mx + b Point-Slope form The equation of the line passing through the point (x₁,y₁) with slope m is : y = m(x - x₁) + y₁ Quadratic Functions and Formulas Examples of Quadratic Functions [Image of a parabola opening up with its vertex at the origin, labeled y = x². The graph is titled "parabola opening up".] [Image of a parabola opening down with its vertex at the origin, labeled y = -x². The graph is titled "parabola opening down".] Forms of Quadratic Functions Standard Form y = ax² + bx + c or f(x) = ax² + bx + c This graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (-b/2a, f(-b/2a)). Vertex Form y = a(x - h)² + k or f(x) = a(x - h)² + k This graph is a parabola that opens up if a > 0 or down if a < 0 and has a vertex at (h,k). Quadratics and Solving for x Quadratic Formula To solve ax² + bx + c = 0, a ≠ 0, use : x = (-b ± √(b² - 4ac))/2a The Discriminant The discriminant is the part of the quadratic equation under the radical, b² - 4ac. We use the discriminant to determine the number of real solutions of ax² + bx + c = 0 as such : If b² - 4ac > 0, there are two real solutions. If b² - 4ac = 0, there is one real solution. If b² - 4ac < 0, there are no real solutions. Square Root Property Let k be a nonnegative number. Then the solutions to the equation x² = k are given by x = ±√k. Other Useful Formulas Compound Interest A = P(1 + r/n)ⁿᵗ where: P= principal of P dollars r= Interest rate (expressed in decimal form) n= number of times compounded per year t= time Continuously Compounded Interest A = Peʳᵗ where: P= principal of P dollars r= Interest rate (expressed in decimal form) t= time Circle (x - h)² + (y - k)² = r² This graph is a circle with radius r and center (h,k). Ellipse (x - h)²/a² + (y - k)²/b² = 1 This is an ellipse with center (h,k) with vertices a units right/left from the center and vertices b units up/down from the center. Hyperbola (x - h)²/a² - (y - k)²/b² = 1 This graph is a hyperbola that opens left and right, has center (h,k), vertices (h ± a,k), and foci (h ± c,k), where c comes from c² = a² + b² and asymptotes that pass through the center y = ± b/a(x - h) + k. (y - k)²/a² - (x - h)²/b² = 1 This graph is a hyperbola that opens up and down, has center (h,k), vertices (h, k ± a), foci (h, k ± c), where c comes from c² = a² + b² and asymptotes that pass through the center y = ± a/b(x - h) + k. Pythagorean Theorem A triangle with legs a and b and hypotenuse c is a right triangle if and only if a² + b² = c² 34.1s