Posted on June 10, 2025 by Mehena Academy
Algebra is the cornerstone of mathematics, providing tools to solve equations, model real-world scenarios, and explore abstract relationships. Whether you're a student preparing for exams, an educator designing lessons, or a math enthusiast, mastering algebra formulas is crucial. This guide offers a comprehensive collection of algebra formulas, complete with explanations, practical examples, and a free PDF download for offline reference. With beautifully rendered equations, this post is your one-stop resource for algebraic success.
Algebra transforms numbers and symbols into meaningful solutions, from calculating distances to optimizing resources. Understanding formulas allows you to simplify expressions, solve equations, and apply mathematics to fields like physics, engineering, and economics. This guide covers formulas across basic algebra, linear equations, quadratics, polynomials, exponents, factoring, sequences, and inequalities, ensuring you have a complete toolkit.
These core formulas underpin algebraic operations, enabling you to manipulate expressions and solve equations efficiently.
Distributive Property: \( a(b + c) = ab + ac \)
Commutative Property: \( a + b = b + a \), \( ab = ba \)
Associative Property: \( (a + b) + c = a + (b + c) \), \( (ab)c = a(bc) \)
Identity Property: \( a + 0 = a \), \( a \cdot 1 = a \)
Inverse Property: \( a + (-a) = 0 \), \( a \cdot \frac{1}{a} = 1 \) (for \( a \neq 0 \))
Example: Simplify \( 3(x + 5) - 2x \).
Apply the distributive property: \( 3(x + 5) = 3x + 15 \). Then, combine like terms: \( 3x + 15 - 2x = x + 15 \).
Tip: Always apply the distributive property to remove parentheses before combining terms.
Linear equations model straight-line relationships, used in graphing, coordinate geometry, and systems of equations.
Standard Form: \( ax + by = c \)
Slope-Intercept Form: \( y = mx + b \), where \( m \) is the slope, \( b \) is the y-intercept
Point-Slope Form: \( y - y_1 = m(x - x_1) \)
Slope Formula: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
Distance Formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)
Midpoint Formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)
Example: Find the equation of the line passing through \( (2, 3) \) and \( (4, 7) \).
Slope: \( m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 \).
Using point-slope form with \( (2, 3) \): \( y - 3 = 2(x - 2) \).
Simplify: \( y - 3 = 2x - 4 \), so \( y = 2x - 1 \).
Tip: Convert to slope-intercept form for easy graphing, as it reveals the slope and y-intercept directly.
Quadratic equations represent parabolas and are solved using factoring, completing the square, or the quadratic formula, with applications in physics and optimization.
Standard Form: \( ax^2 + bx + c = 0 \)
Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Discriminant: \( \Delta = b^2 - 4ac \)
Sum of Roots: \( \frac{-b}{a} \)
Product of Roots: \( \frac{c}{a} \)
Vertex Form: \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex
Example: Solve \( 3x^2 + 6x - 9 = 0 \).
Here, \( a = 3 \), \( b = 6 \), \( c = -9 \). Discriminant: \( \Delta = 6^2 - 4(3)(-9) = 36 + 108 = 144 \).
Quadratic formula: \( x = \frac{-6 \pm \sqrt{144}}{2(3)} = \frac{-6 \pm 12}{6} \).
Solutions: \( x = \frac{-6 + 12}{6} = 1 \), \( x = \frac{-6 - 12}{6} = -3 \).
Verify: Sum of roots: \( 1 + (-3) = -2 = \frac{-6}{3} \). Product: \( 1 \cdot (-3) = -3 = \frac{-9}{3} \).
Tip: A positive discriminant indicates two real roots, zero indicates one, and negative indicates no real roots.
Exponents handle powers and roots, while polynomials extend algebraic expressions, critical in calculus and modeling.
Product Rule: \( a^m \cdot a^n = a^{m+n} \)
Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \)
Power Rule: \( (a^m)^n = a^{mn} \)
Zero Exponent: \( a^0 = 1 \) (for \( a \neq 0 \))
Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
Fractional Exponent: \( a^{\frac{1}{n}} = \sqrt[n]{a} \)
Polynomial Multiplication: \( (ax + b)(cx + d) = acx^2 + (ad + bc)x + bd \)
Example: Simplify \( \frac{x^6}{x^2} \cdot x^4 \).
Quotient rule: \( \frac{x^6}{x^2} = x^{6-2} = x^4 \). Then, \( x^4 \cdot x^4 = x^{4+4} = x^8 \).
Example: Multiply \( (3x + 2)(x - 4) \).
Polynomial multiplication: \( (3x \cdot x) + (3x \cdot -4) + (2 \cdot x) + (2 \cdot -4) = 3x^2 - 12x + 2x - 8 = 3x^2 - 10x - 8 \).
Factoring decomposes expressions into simpler factors, facilitating equation solving and simplification.
Difference of Squares: \( a^2 - b^2 = (a - b)(a + b) \)
Perfect Square Trinomial: \( a^2 + 2ab + b^2 = (a + b)^2 \), \( a^2 - 2ab + b^2 = (a - b)^2 \)
Sum of Cubes: \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \)
Difference of Cubes: \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \)
Quadratic Trinomial: \( x^2 + (a + b)x + ab = (x + a)(x + b) \)
Example: Factor \( x^2 - 25 \).
Difference of squares: \( x^2 - 25 = x^2 - 5^2 = (x - 5)(x + 5) \).
Example: Factor \( x^2 + 6x + 8 \).
Factors of 8 summing to 6: \( 2 \cdot 4 = 8 \), \( 2 + 4 = 6 \). Thus, \( x^2 + 6x + 8 = (x + 2)(x + 4) \).
Sequences model ordered patterns, used in finance, computer science, and statistics.
Arithmetic Sequence: \( a_n = a_1 + (n - 1)d \), where \( d \) is the common difference
Sum of Arithmetic Series: \( S_n = \frac{n}{2}(a_1 + a_n) \) or \( S_n = \frac{n}{2}[2a_1 + (n - 1)d] \)
Geometric Sequence: \( a_n = a_1 r^{n-1} \), where \( r \) is the common ratio
Sum of Geometric Series: \( S_n = a_1 \frac{1 - r^n}{1 - r} \) (for \( r \neq 1 \))
Infinite Geometric Series: \( S = \frac{a_1}{1 - r} \) (for \( |r| < 1 \))
Example: Find the 8th term of the geometric sequence \( 2, 6, 18, \ldots \).
Common ratio: \( r = \frac{6}{2} = 3 \). First term: \( a_1 = 2 \). Formula: \( a_8 = 2 \cdot 3^{8-1} = 2 \cdot 3^7 = 2 \cdot 2187 = 4374 \).
Inequalities define solution ranges, requiring careful handling of inequality signs.
Linear Inequality: \( ax + b > c \) (solve as equation, adjust sign if multiplying/dividing by negative)
Quadratic Inequality: \( ax^2 + bx + c > 0 \) (solve \( ax^2 + bx + c = 0 \), test intervals)
Absolute Value Inequality: \( |x - a| < b \implies a - b < x < a + b \), \( |x - a| > b \implies x < a - b \text{ or } x > a + b \)
Example: Solve \( -3x + 6 \leq 12 \).
Subtract 6: \( -3x \leq 6 \). Divide by -3 (flip sign): \( x \geq -2 \).
Example: Solve \( |x + 2| > 5 \).
Absolute value rule: \( x + 2 < -5 \text{ or } x + 2 > 5 \). Solve: \( x < -7 \text{ or } x > 3 \).
Tip: Graph quadratic inequalities on a number line to visualize solution intervals.
With these algebra formulas, you're equipped to solve a wide range of problems. Download the PDF for a handy reference and visit our Maths Tools to practice with tools like the Algebra Equation Solver. Explore more tips in our Short Tricks for Multiplication post.