December 27, 2024 , Frantz

Most important and commonly used algebraic formulas

1. Basic Arithmetic Formulas

  • Addition and Subtraction:
    • \( a + b = b + a \) (Commutative property)
  • Multiplication and Division:
    • \( a \times b = b \times a \) (Commutative property)
    • \( \frac{a}{b} \) (Quotient)

2. Exponents and Powers

  • Power of a product:
    \[
    (ab)^n = a^n \times b^n
    \]
  • Power of a quotient:
    \[
    \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
    \]
  • Zero exponent:
    \[
    a^0 = 1 \quad (\text{for any non-zero } a)
    \]
  • Negative exponent:
    \[
    a^{-n} = \frac{1}{a^n}
    \]
  • Fractional exponent:
    \[
    a^{\frac{m}{n}} = \sqrt[n]{a^m}
    \]

3. Polynomials

  • Quadratic Formula:
    \[
    x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}
    \]
    (for the equation \( ax^2 + bx + c = 0 \))
  • Factoring of Quadratics:
    \[
    ax^2 + bx + c = (px + q)(rx + s)
    \]
  • Difference of Squares:
    \[
    a^2 – b^2 = (a + b)(a – b)
    \]
  • Perfect Square Trinomial:
    \[
    a^2 + 2ab + b^2 = (a + b)^2
    \]
  • Sum and Difference of Cubes:
    \[
    a^3 + b^3 = (a + b)(a^2 – ab + b^2)
    \]
    \[
    a^3 – b^3 = (a – b)(a^2 + ab + b^2)
    \]

4. Linear Equations

  • Slope of a line:
    \[
    m = \frac{y_2 – y_1}{x_2 – x_1}
    \]
  • Point-slope form:
    \[
    y – y_1 = m(x – x_1)
    \]
  • Slope-intercept form:
    \[
    y = mx + b
    \]
  • Standard form of a line:
    \[
    Ax + By = C
    \]

5. Systems of Equations

  • Substitution method: Solve one equation for one variable, then substitute it into the other.
  • Elimination method: Add or subtract the equations to eliminate one variable.
  • Cramer’s Rule: For a system of two equations:
    \[
    \frac{x}{y} = \frac{det(A_x)}{det(A)}
    \]

6. Inequalities

  • Linear inequality:
    \[
    ax + b > c \quad \text{or} \quad \neq, \leq, \geq
    \]
  • Multiplying/dividing by a negative number: If you multiply or divide an inequality by a negative number, flip the inequality sign.
    \[
    \text{If } \quad a < b \quad \text{ and } \quad c < 0, \quad \text{ then } \quad ac > bc
    \]

    • 7. Logarithms

    • Logarithmic identity:
      \[
      \log_b(xy) = \log_b(x) + \log_b(y)
      \]
      \[
      \log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y)
      \]
    • Power rule:
      \[
      \log_b(x^n) = n \cdot \log_b(x)
      \]
    • Change of base formula:
      \[
      \log_b(x) = \frac{\log_k(x)}{\log_k(b)}
      \]
    • Logarithmic form of exponentiation:
      \[
      b^x = y \quad \text{is equivalent to} \quad \log_b(y) = x
      \]

    8. Rational Expressions

    • Addition/Subtraction of fractions:
      \[
      \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
      \]
    • Multiplication of fractions:
      \[
      \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
      \]
    • Division of fractions:
      \[
      \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
      \]

    9. Functions

    • Function notation:
      \[
      f(x) = mx + b \quad (\text{linear function})
      \]
    • Composite functions:
      \[
      (f \circ g)(x) = f(g(x))
      \]

    10. Sequences and Series

    • Arithmetic sequence:
      \[
      a_n = a_1 + (n-1) \cdot d
      \]

7. Logarithms

  • Logarithmic identity:
    \[
    \log_b(xy) = \log_b(x) + \log_b(y)
    \]
    \[
    \log_b\left(\frac{x}{y}\right) = \log_b(x) – \log_b(y)
    \]
  • Power rule:
    \[
    \log_b(x^n) = n \cdot \log_b(x)
    \]

  • Change of base formula:
    \[
    \log_b(x) = \frac{\log_k(x)}{\log_k(b)}
    \]
  • Logarithmic form of exponentiation:
    \[
    b^x = y \quad \text{is equivalent to} \quad \log_b(y) = x
    \]

    • 8. Rational Expressions

    • Addition/Subtraction of fractions:
      \[
      \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
      \]
    • Multiplication of fractions:
      \[
      \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
      \]
    • Division of fractions:
      \[
      \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}
      \]

    9. Functions

    • Function notation:
      \[
      f(x) = mx + b \quad (\text{linear function})
      \]
    • Composite functions:
      \[
      (f \circ g)(x) = f(g(x))
      \]

    10. Sequences and Series

    • Arithmetic sequence:
      \[
      a_n = a_1 + (n-1) \cdot d
      \]
    • Sum of an arithmetic series:
      \[
      S_n = \frac{n}{2} \cdot (a_1 + a_n)
      \]
    • Geometric sequence:
      \[
      a_n = a_1 \cdot r^{n-1}
      \]
    • Sum of a geometric series:
      \[
      S_n = a_1 \cdot \frac{1 – r^n}{1 – r} \quad (\text{for} \ r \neq 1)
      \]

    11. Quadratic Functions

    • Vertex form of a quadratic equation:
      \[
      y = a(x – h)^2 + k
      \]
    • Standard form:
      \[
      y = ax^2 + bx + c
      \]
    • Discriminant:
      \[
      D = b^2 – 4ac
      \]

      • If \( D > 0 \), Two real roots
      • If \( D = 0 \), One real root
      • If \( D < 0 \), no real roots

    12. Binomial Theorem

    • Expansion of \((a + b)^n\):
      \[
      (a + b)^n = \sum_{k=0}^{n} \left( \begin{array}{c} n \\ k \end{array} \right) a^{n-k} b^k
      \]
      \[
      \text{Where} \quad \left( \begin{array}{c} n \\ k \end{array} \right) \quad \text{is the binomial coefficient.}
      \]

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