📐 Math ResourcesLast updated May 3, 2026

Algebra & Equations Guide 2026: Solving Linear, Quadratic, Systems, Polynomials & More

Step-by-step algebra from basic equations to advanced polynomial and matrix methods

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Free Calculators

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20+

Topics Covered

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Equation Types

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15 min

Read Time

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Key Takeaways

Order of operations (PEMDAS/BODMAS): Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right) — always evaluate in this order.
Linear equation solving: isolate the variable by performing inverse operations on both sides (add/subtract, then multiply/divide).
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a — solves ax² + bx + c = 0. The discriminant (b² − 4ac) tells you whether there are 2 real, 1 real, or 0 real roots.
Exponent rules: aᵐ × aⁿ = aᵐⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ; (ab)ⁿ = aⁿ × bⁿ.
Logarithms are inverse exponents: logᵇ(x) = y ⟺ bʸ = x. Product rule: log(ab) = log(a) + log(b); Quotient: log(a/b) = log(a) − log(b); Power: log(aⁿ) = n⋅log(a).
FOIL method for binomials: (a + b)(c + d) = ac + ad + bc + bd — First, Outer, Inner, Last.
Systems of equations — three methods: substitution (solve one equation for one variable, substitute), elimination (add/subtract equations to cancel a variable), matrices (Gaussian elimination or Cramer’s rule).
To factor ax² + bx + c: find two numbers that multiply to ac and add to b, then split the middle term and factor by grouping.
Slope-intercept form: y = mx + b (m = slope = rise/run; b = y-intercept). Standard form: Ax + By = C.
Binary ↔ Decimal: Group binary digits in 4s from right; each position = 2ⁿ. Hex ↔ Binary: Each hex digit maps to exactly 4 binary digits (0–9 and A–F = 0000–1111).

Algebra is the mathematical language of patterns, relationships, and unknown quantities. From balancing a budget to designing circuits, algebraic thinking underpins virtually every quantitative field. This 2026 guide — aligned with Khan Academy Algebra, AP Algebra curriculum, and Purplemath standards — covers every foundational and intermediate algebra topic with worked examples and our free calculators including the equation solver, fraction calculator, and matrix calculator.

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Order of Operations: PEMDAS / BODMAS Explained

Before solving any algebraic expression, you must evaluate operations in the correct order. PEMDAS (US) / BODMAS (UK): Parentheses (brackets) → Exponents (orders) → Multiplication & Division (left to right) → Addition & Subtraction (left to right). Example: 3 + 6 × (5 + 4) ÷ 3 − 7 = 3 + 6 × 9 ÷ 3 − 7 = 3 + 54 ÷ 3 − 7 = 3 + 18 − 7 = 14. Brackets override all other rules. Multiplication and division have equal priority — evaluate left to right. Same for addition and subtraction. A common mistake: computing 8 ÷ 2(2+2) = 8 ÷ 2 × 4 = 16, not 1 — because ÷ and × are evaluated left to right.

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Linear Equations & Inequalities: Solving Step by Step

A linear equation has degree 1 (no x² or higher). Standard form: ax + b = c. Solving process: Distribute → combine like terms → move variables to one side → move constants to other side → divide by coefficient. Special cases: No solution when you get a false statement (3 = 5); infinitely many solutions when you get a true statement (0 = 0). Linear inequalities work the same but flip the inequality sign when multiplying or dividing by a negative number. Example: −3x < 12 → x > −4 (sign flips). Slope-intercept form: y = mx + b — m is the slope (rise over run), b is the y-intercept. Finding slope from two points: m = (y₂ − y₁) / (x₂ − x₁). Use our equation solver for instant solutions.

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Quadratic Equations: Factoring, Completing the Square & Quadratic Formula

A quadratic equation has degree 2: ax² + bx + c = 0. Three solving methods: (1) Factoring: Rewrite as product of two binomials — fastest when possible. (2) Completing the square: Convert to (x+h)² = k form — useful for deriving other properties. (3) Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a — always works. The discriminant D = b² − 4ac: D > 0 → two real roots; D = 0 → one real root; D < 0 → two complex roots. Vertex form: y = a(x−h)² + k — vertex at (h, k), axis of symmetry x = h. Sum and product of roots: x₁ + x₂ = −b/a; x₁ × x₂ = c/a (Viète’s formulas).

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Systems of Equations: Substitution, Elimination & Matrices

A system of equations involves two or more equations with two or more unknowns. Graphical interpretation: The solution is the point(s) where lines intersect. Exactly one solution = lines cross; no solution = parallel lines; infinite solutions = same line. Substitution method: Solve one equation for one variable, substitute into the other. Best when a coefficient is 1. Elimination method: Multiply equations by constants to make coefficients equal, then add/subtract to eliminate one variable. Matrix method: Write as an augmented matrix [A|b] and row-reduce (Gaussian elimination). For a 2×2 system: use Cramer’s rule (x = Dₓ/D, y = Dᵧ/D where D is the determinant). Systems with 3 variables require either elimination chains or matrices. Use our equation solver for multi-variable systems.

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Polynomials & Factoring: FOIL, Grouping & Special Products

A polynomial is a sum of terms with non-negative integer exponents: aₙxⁿ + ... + a₁x + a₀. FOIL method multiplies two binomials: (a+b)(c+d) = ac + ad + bc + bd. Special products: (a+b)² = a² + 2ab + b²; (a−b)² = a² − 2ab + b²; (a+b)(a−b) = a² − b² (difference of squares). Factoring strategies: (1) Always factor out the GCF first. (2) Difference of squares: a² − b² = (a+b)(a−b). (3) Perfect square trinomials: a² ± 2ab + b² = (a ± b)². (4) Sum/difference of cubes: a³ + b³ = (a+b)(a² − ab + b²). (5) Factor by grouping for 4+ terms. Polynomial long division divides polynomials just like numeric long division; synthetic division works for linear divisors (x − k).

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Exponents & Radical Expressions: All Rules with Examples

The 7 exponent rules: Product (aᵐ·aⁿ = aᵐⁿⁿ), Quotient (aᵐ/aⁿ = aᵐ⁻ⁿ), Power of power ((aᵐ)ⁿ = aᵐⁿⁿ), Power of product ((ab)ⁿ = aⁿbⁿ), Zero (a⁰ = 1), Negative (a⁻ⁿ = 1/aⁿ), Fractional (a^(m/n) = ⁿ√aᵐ). Simplifying radicals: √(48) = √(16×3) = 4√3. Rationalize denominators: multiply by conjugate (1/√2 = √2/2). Scientific notation: 6.02 × 10²³ — move decimal point. Multiplying: (3×10²) × (4×10³) = 12×10⁵ = 1.2×10⁶. Use our exponent calculator for any base and power computation.

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Logarithms: Properties, Change of Base & Applications

A logarithm answers: "to what exponent must I raise the base to get this value?" logᵇ(x) = y means bʸ = x. Common bases: log₁₀ (common, base 10), ln (natural, base e ≈ 2.71828), log₂ (binary, used in computing and information theory). Three core properties: Product: log(MN) = log(M) + log(N); Quotient: log(M/N) = log(M) − log(N); Power: log(Mⁿ) = n⋅log(M). Change of base formula: logᵇ(x) = log(x)/log(b) = ln(x)/ln(b). Real-world applications: pH = −log[H⁺] (chemistry); Sound dB = 10⋅log₁₀(I/I₀) (acoustics); Richter magnitude = log₁₀(A/A₀) (earthquakes); Compound interest: t = ln(A/P) / (r⋅ln(e)) = ln(A/P)/r. Use our logarithm calculator for any base.

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Fractions & Rational Expressions: Arithmetic and Simplification

Fraction arithmetic: Add/subtract — find LCD, convert, add/subtract numerators, simplify. Multiply: multiply numerators × numerators, denominators × denominators; cancel common factors before multiplying. Divide: multiply by reciprocal (flip the divisor). Simplifying: Divide numerator and denominator by their GCD. Example: 18/24 → GCD(18,24) = 6 → 3/4. Rational expressions are fractions with polynomials. Simplify by factoring numerator and denominator, then cancel common factors. Example: (x²−9)/(x+3) = (x+3)(x−3)/(x+3) = x−3 (x ≠ −3). To add rational expressions: find LCD (LCM of denominators), convert, add numerators. Always state restrictions (values where denominator = 0). Use our fraction calculator for instant fraction arithmetic.

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Absolute Value Equations & Inequalities

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Matrices & Number Systems: Operations and Conversions

Matrix operations: Addition/subtraction require same dimensions — add corresponding elements. Multiplication: A (m×n) × B (n×p) = C (m×p); columns of A must equal rows of B. Determinant of 2×2: |A| = ad − bc for [[a,b],[c,d]]. Inverse of 2×2: A⁻¹ = (1/|A|) × [[d,−b],[−c,a]]. Matrix equation AX = B → X = A⁻¹B. Number systems: Decimal (base 10), Binary (base 2, used in computing), Octal (base 8), Hexadecimal (base 16, uses A–F). Conversion shortcuts: Each hex digit = 4 binary digits. Binary 1111 = Hex F = Decimal 15. Bitwise operations (AND, OR, XOR, NOT) operate on binary representations. Use our matrix calculator and binary calculator.

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Product, quotient, power rules with worked examples and real-world applications.

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Frequently Asked Questions

How do you solve a linear equation step by step?

Solving a linear equation (ax + b = c): Step 1: Distribute any parentheses on both sides. Step 2: Combine like terms on each side. Step 3: Move all variable terms to one side using addition or subtraction. Step 4: Move all constant terms to the other side. Step 5: Divide both sides by the coefficient of the variable. Example: 3x + 7 = 16 → 3x = 9 → x = 3. Check: 3(3) + 7 = 16 ✓. Always verify your answer by substituting back into the original equation.

What is the quadratic formula and when do you use it?

The quadratic formula solves any equation in the form ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / 2a. Use it when the quadratic cannot be easily factored. The discriminant D = b² − 4ac determines the solutions: D > 0 means two distinct real roots; D = 0 means one repeated real root (vertex touches x-axis); D < 0 means no real roots (two complex roots). Example: x² − 5x + 6 = 0 → a=1, b=−5, c=6 → x = (5 ± √(25−24))/2 = (5 ± 1)/2 → x = 3 or x = 2.

What is the difference between substitution and elimination for systems of equations?

Substitution: Solve one equation for one variable, then substitute that expression into the other equation. Best when one variable has coefficient 1. Example: y = 2x + 1; 3x + y = 10 → 3x + (2x+1) = 10 → 5x = 9. Elimination: Add or subtract equations to cancel one variable. Multiply equations by constants first if needed. Best when coefficients are multiples. Both methods yield the same solution. Use matrices (Gaussian elimination) for 3+ variables.

How do you factor a trinomial ax² + bx + c?

For x² + bx + c (a=1): Find two numbers p and q where p×q = c and p+q = b. Then factor as (x+p)(x+q). Example: x² + 5x + 6 → need 2×3=6 and 2+3=5 → (x+2)(x+3). For ax² + bx + c (a≠1): Multiply a×c, find factors of ac that add to b, split the middle term, then factor by grouping. Example: 2x² + 7x + 3 → a×c = 6; factors 6×1=6, 6+1=7 → 2x² + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3).

How do you add and subtract fractions with different denominators?

Step 1: Find the Least Common Denominator (LCD) of the two fractions. Step 2: Convert each fraction to an equivalent fraction with the LCD. Step 3: Add or subtract the numerators; keep the denominator. Step 4: Simplify by dividing by GCD. Example: 2/3 + 3/4 → LCD = 12 → 8/12 + 9/12 = 17/12. For subtraction: 5/6 − 1/4 → LCD = 12 → 10/12 − 3/12 = 7/12. Use our fraction calculator for any operation with automatic simplification.

What are all the exponent rules?

The seven essential exponent rules: (1) Product: aᵐ×aⁿ = aᵐⁿⁿ. (2) Quotient: aᵐ/aⁿ = aᵐ⁻ⁿ. (3) Power of power: (aᵐ)ⁿ = aᵐ·ⁿ. (4) Power of product: (ab)ⁿ = aⁿbⁿ. (5) Power of quotient: (a/b)ⁿ = aⁿ/bⁿ. (6) Zero exponent: a⁰ = 1 (a≠0). (7) Negative exponent: a⁻ⁿ = 1/aⁿ. Fractional exponents: a^(m/n) = ⁿ√aᵐ = (ⁿ√a)ᵐ. Example: 8^(2/3) = (³√8)² = 2² = 4.

What is completing the square and why is it used?

Completing the square transforms ax² + bx + c into a(x + h)² + k form. Steps for x² + bx + c = 0: (1) Move c: x² + bx = −c. (2) Add (b/2)² to both sides: x² + bx + (b/2)² = −c + (b/2)². (3) Factor left side: (x + b/2)² = right side. (4) Square root both sides. Uses: Deriving the quadratic formula, finding the vertex of a parabola, solving quadratics that are hard to factor, graphing analysis (vertex form y = a(x−h)² + k tells you vertex (h, k) immediately).

How do logarithm properties work?

The three core logarithm properties: Product rule: logᵇ(MN) = logᵇ(M) + logᵇ(N). Quotient rule: logᵇ(M/N) = logᵇ(M) − logᵇ(N). Power rule: logᵇ(Mⁿ) = n⋅logᵇ(M). Change of base: logᵇ(x) = ln(x)/ln(b) = log(x)/log(b). Common log: log₁₀(10000) = 4. Natural log: ln(e³) = 3. Applications: pH = −log[H⁺] (chemistry); decibels dB = 10⋅log(P/P₀) (acoustics); Richter scale (seismology); half-life/compound interest formulas.

How do you solve absolute value equations?

|x| = a has two solutions: x = a or x = −a (when a ≥ 0). For |ax + b| = c: Set up two equations: ax + b = c AND ax + b = −c. Solve each separately. Check both solutions in the original equation. For |ax + b| = |cx + d|: Set up ax + b = cx + d OR ax + b = −(cx + d). For inequalities: |x| < a means −a < x < a (AND); |x| > a means x < −a OR x > a (OR). Example: |2x − 3| = 7 → 2x−3 = 7 (x=5) or 2x−3 = −7 (x=−2).

How does matrix multiplication work?

For matrix A (m×n) times B (n×p), the result is C (m×p). Each element C[i][j] = sum of A[i][k] × B[k][j] for k=1 to n. The number of columns in A must equal the number of rows in B — otherwise multiplication is undefined. Example: [1 2; 3 4] × [5 6; 7 8] = [1×5+2×7, 1×6+2×8; 3×5+4×7, 3×6+4×8] = [19 22; 43 50]. Note: matrix multiplication is NOT commutative — AB ≠ BA in general. Use our matrix calculator for 2×2 through 4×4 operations.

How do you convert between binary, decimal, and hexadecimal?

Decimal to Binary: Divide by 2 repeatedly, record remainders bottom-up. Example: 25 → 11001. Binary to Decimal: Multiply each digit by 2^n (right to left from n=0). Example: 11001 = 16+8+0+0+1 = 25. Decimal to Hex: Divide by 16, use A–F for 10–15. Example: 255 = FF. Hex to Binary: Replace each hex digit with its 4-bit binary equivalent. Example: 2F = 0010 1111. Binary to Hex: Group into blocks of 4 from right. Example: 101111 = 0010 1111 = 2F.

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