Complete K-12 Algebraic Equations

🎯 Complete K-12 Algebraic Equations 🎯

Master Every Type of Equation from Elementary to High School

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📑 Equation Types & Topics

1. Linear Equations 2. Quadratic Equations 3. Systems of Equations 4. Polynomial Equations 5. Rational Equations 6. Radical Equations 7. Exponential Equations 8. Absolute Value Equations 9. Special Equations

1. Linear Equations (Grade 6-9)

Standard Form

ax + b = c

ax + by = c

Definition: An equation where the highest power of the variable is 1.
Forms a straight line when graphed.

Solving Methods

One-Step Equations

x + a = b → x = b − a

x − a = b → x = b + a

ax = b → x = ba

xa = b → x = ab

Two-Step Equations

1 Isolate the term with variable (add/subtract)

2 Isolate the variable (multiply/divide)

Example: Solve 3x + 7 = 22

3x + 7 = 22

3x = 22 − 7

3x = 15

x = 5

Multi-Step Equations

1 Distribute if needed

2 Combine like terms on each side

3 Move variable terms to one side

4 Move constant terms to other side

5 Solve for the variable

Example: Solve 2(x − 3) + 5 = 4x − 1

2(x − 3) + 5 = 4x − 1

2x − 6 + 5 = 4x − 1

2x − 1 = 4x − 1

2x − 4x = −1 + 1

−2x = 0

x = 0

Equations with Fractions

xa + xb = c

Method: Multiply everything by the LCD (Least Common Denominator)

Example: Solve x3 + x4 = 14

Multiply by LCD = 12

12 × (x3) + 12 × (x4) = 12 × 14

4x + 3x = 168

7x = 168

x = 24

Special Cases

Identity (Infinite Solutions)

2x + 3 = 2x + 3

True for all values of x

Contradiction (No Solution)

x + 5 = x + 8

No value of x satisfies this

2. Quadratic Equations (Grade 9-11)

Standard Form

ax² + bx + c = 0

Definition: An equation where the highest power is 2.
a ≠ 0 (coefficient of x² must not be zero)

Solving Methods

Method 1: Factoring

x² + bx + c = (x + m)(x + n) = 0

Find m and n where:
• m + n = b
• m × n = c

Example: Solve x² + 7x + 12 = 0

Need: m + n = 7 and m × n = 12

Numbers: 3 and 4

(x + 3)(x + 4) = 0

x + 3 = 0 or x + 4 = 0

x = −3 or x = −4

Method 2: Quadratic Formula

x = −b ± b² − 4ac 2a

Works for ALL quadratic equations!

Example: Solve 2x² + 5x − 3 = 0

a = 2, b = 5, c = −3

x = −5 ± 25−4(2)(−3)4

x = −5 ± 494

x = −5 ± 74

x = 24 = 12 or x = −124 = −3

Method 3: Completing the Square

x² + bx = c

x² + bx + (b2)² = c + (b2)²

Example: Solve x² + 6x − 7 = 0

x² + 6x = 7

x² + 6x + 9 = 7 + 9

(x + 3)² = 16

x + 3 = ±4

x = 1 or x = −7

Method 4: Square Root Method

x² = k → x = ±k

(x − h)² = k → x − h = ±k

The Discriminant

Δ = b² − 4ac

Discriminant	Number of Solutions	Type
Δ > 0	2 real solutions	Two different values
Δ = 0	1 real solution	Repeated root
Δ < 0	0 real solutions	Complex numbers

Sum and Product of Roots

Sum: r₁ + r₂ = −ba

Product: r₁ × r₂ = ca

3. Systems of Equations (Grade 8-10)

Types of Systems

System of equations: Two or more equations with same variables solved simultaneously

Linear Systems (2 Variables)

{a₁x + b₁y = c₁
a₂x + b₂y = c₂}

Method 1: Substitution

1 Solve one equation for one variable

2 Substitute into the other equation

3 Solve for the remaining variable

4 Back-substitute to find the other variable

Example: Solve {x + 2y = 7 and 3x − y = 5}

From first: x = 7 − 2y

Substitute: 3(7 − 2y) − y = 5

21 − 6y − y = 5

21 − 7y = 5

−7y = −16

y = 167

x = 7 − 2(167) = 177

Method 2: Elimination (Addition/Subtraction)

1 Multiply equations to get equal coefficients

2 Add or subtract to eliminate one variable

3 Solve for remaining variable

4 Substitute back to find other variable

Example: Solve {2x + 3y = 12 and 4x − y = 10}

Multiply second by 3: 12x − 3y = 30

Add to first: 2x + 3y = 12

14x = 42

x = 3

Substitute: 2(3) + 3y = 12

6 + 3y = 12

y = 2

Method 3: Graphing

Solution is the point where lines intersect

Types of Solutions

One Solution

Lines intersect

Consistent & Independent

No Solution

Parallel lines

Inconsistent

Infinite Solutions

Same line

Consistent & Dependent

3-Variable Systems

{a₁x + b₁y + c₁z = d₁
a₂x + b₂y + c₂z = d₂
a₃x + b₃y + c₃z = d₃}

Solve using elimination to reduce to 2-variable system

Non-Linear Systems

{y = x² + 2x − 3
y = 2x + 1}

One or both equations is non-linear (quadratic, etc.)

4. Polynomial Equations (Grade 10-12)

General Form

a_nxⁿ + a_n-1x^n-1 + ... + a₁x + a₀ = 0

Degree: Highest power of x
A polynomial of degree n has at most n real roots

Cubic Equations (Degree 3)

ax³ + bx² + cx + d = 0

Factoring Method

Look for common factors or use grouping

Example: Solve x³ − 6x² + 11x − 6 = 0

Try x = 1: 1 − 6 + 11 − 6 = 0 ✓

(x − 1) is a factor

x³ − 6x² + 11x − 6 = (x − 1)(x² − 5x + 6)

(x − 1)(x − 2)(x − 3) = 0

x = 1, 2, or 3

Quartic Equations (Degree 4)

ax⁴ + bx³ + cx² + dx + e = 0

Special Type: Biquadratic

ax⁴ + bx² + c = 0

Method: Let u = x², then solve au² + bu + c = 0

Example: Solve x⁴ − 5x² + 4 = 0

Let u = x²

u² − 5u + 4 = 0

(u − 1)(u − 4) = 0

u = 1 or u = 4

x² = 1 or x² = 4

x = ±1 or x = ±2

Important Theorems

Remainder Theorem

f(a) = remainder when f(x) ÷ (x − a)

Factor Theorem

If f(a) = 0, then (x − a) is a factor of f(x)

Rational Root Theorem

Possible rational roots = ±factors of constantfactors of leading coefficient

Synthetic Division

Quick method to divide polynomial by (x − a)
Used to test possible roots and factor polynomials

Sum and Product of Roots

For axⁿ + bx^n-1 + ... = 0

Sum of roots = −ba

Product of roots = (−1)ⁿ constant terma

5. Rational Equations (Grade 10-11)

Definition

P(x)Q(x) = R(x)S(x)

Rational equation: Contains fractions with variables in denominator
Restriction: Denominator cannot be zero!

Solving Method

1 Find restrictions (values that make denominator = 0)

2 Find LCD of all denominators

3 Multiply entire equation by LCD

4 Solve resulting polynomial equation

5 Check solutions (reject any that violate restrictions)

Example: Solve 3x−2 + 1x+1 = 2x²−x−2

Restrictions: x ≠ 2, x ≠ −1

Factor: x² − x − 2 = (x−2)(x+1)

LCD = (x−2)(x+1)

3(x+1) + 1(x−2) = 2

3x + 3 + x − 2 = 2

4x + 1 = 2

4x = 1

x = 14 ✓ (Valid)

Cross Multiplication

If ab = cd, then ad = bc

Example: Solve 2x+1x−3 = 52

Cross multiply: 2(2x+1) = 5(x−3)

4x + 2 = 5x − 15

2 + 15 = 5x − 4x

x = 17

Work Rate Problems

1time₁ + 1time₂ = 1time together

Extraneous Solutions

Warning: Always check solutions!
Solutions that make any denominator = 0 must be rejected

6. Radical Equations (Grade 10-11)

Definition

expression = value

Radical equation: Contains variable under a radical (square root, cube root, etc.)

Solving Square Root Equations

1 Isolate the radical on one side

2 Square both sides

3 Solve resulting equation

4 Check all solutions (squaring can introduce extraneous solutions)

Example: Solve 2x + 3 = 5

2x + 3 = 5

Square both sides: 2x + 3 = 25

2x = 22

x = 11

Check: 2(11) + 3 = 25 = 5 ✓

Equations with Two Radicals

Example: Solve x + 7 = x − 5

Square both sides: x + 7 = x − 5

7 = −5 (False!)

No solution

Equations Requiring Multiple Squarings

x + x−5 = 5

1 Isolate one radical

2 Square both sides

3 Isolate remaining radical

4 Square again and solve

Cube Root Equations

∛x = a → x = a³

Cube both sides (does not introduce extraneous solutions)

Example: Solve ∛(2x − 1) = 3

Cube both sides: 2x − 1 = 27

2x = 28

x = 14

Power Equations

x^m/n = k

x = k^n/m

💡 Important Tips

• Always check solutions when squaring

• Square root of negative numbers is not real

• Even roots require non-negative radicands

7. Exponential & Logarithmic Equations (Grade 11-12)

Exponential Equations

a^x = b

Definition: Variable is in the exponent

Method 1: Same Base

If a^x = a^y, then x = y

Example: Solve 2^x+1 = 8

2^x+1 = 2³

x + 1 = 3

x = 2

Method 2: Using Logarithms

a^x = b → x = log_ab

x = log blog a or x = ln bln a

Example: Solve 3^x = 20

Take log of both sides: log(3^x) = log(20)

x log 3 = log 20

x = log 20log 3

x ≈ 2.727

Logarithmic Equations

log_ax = b → x = a^b

Method 1: Convert to Exponential

Example: Solve log₂(x) = 5

x = 2⁵

x = 32

Method 2: Use Log Properties

log(a) + log(b) = log(ab)

log(a) − log(b) = log(ab)

n log(a) = log(aⁿ)

Example: Solve log(x) + log(x−3) = 1

log[x(x−3)] = 1

x(x−3) = 10¹

x² − 3x = 10

x² − 3x − 10 = 0

(x − 5)(x + 2) = 0

x = 5 or x = −2

Check: x = 5 ✓ (x = −2 invalid, log of negative)

Natural Logarithm Equations

ln(x) = a → x = e^a

e^x = b → x = ln(b)

Change of Base Formula

log_ab = log_cblog_ca

💡 Important Restrictions

• log_ax is only defined for x > 0

• Base a must be positive and a ≠ 1

• Always check solutions

8. Absolute Value Equations (Grade 9-10)

Definition

|x| = {x if x ≥ 0; −x if x < 0}

Absolute value: Distance from zero (always non-negative)

Basic Form

|x| = a

x = a or x = −a (if a ≥ 0)

If a < 0, there is no solution (absolute value cannot be negative)

Example: Solve |x| = 7

x = 7 or x = −7

General Form

|ax + b| = c

ax + b = c or ax + b = −c

Example: Solve |2x − 5| = 9

Case 1: 2x − 5 = 9

2x = 14

x = 7

Case 2: 2x − 5 = −9

2x = −4

x = −2

Solutions: x = 7 or x = −2

Absolute Value Equal to Expression

|f(x)| = |g(x)|

f(x) = g(x) or f(x) = −g(x)

Example: Solve |x − 3| = |2x + 1|

Case 1: x − 3 = 2x + 1

−3 − 1 = 2x − x

x = −4

Case 2: x − 3 = −(2x + 1)

x − 3 = −2x − 1

3x = 2

x = 23

Multiple Absolute Values

|a| + |b| = c

Break into cases based on signs of expressions inside absolute values

Special Cases

No Solution

|x| = −5

Absolute value cannot be negative

One Solution

|x| = 0

x = 0 (only value with absolute value 0)

💡 Solving Strategy

1. Isolate the absolute value expression

2. Check if right side is non-negative

3. Split into two cases (positive and negative)

4. Solve both cases

5. Check all solutions

9. Special Types of Equations (Grade 11-12)

Literal Equations

Literal equation: Equation with multiple variables; solve for one in terms of others

Example: Solve for r in A = πr²

A = πr²

Aπ = r²

r = ±Aπ

Parametric Equations

{x = f(t)
y = g(t)}

Both x and y defined in terms of parameter t

Piecewise Equations

Function defined differently on different intervals

f(x) = {x² if x < 0
2x if x ≥ 0}

Trigonometric Equations

sin(x) = a

cos(x) = b

tan(x) = c

Usually have infinite solutions (periodic)
Find solutions in given interval

Example: Solve sin(x) = 12 for 0 ≤ x ≤ 2π

x = π6 or x = 5π6

Matrix Equations

AX = B

X = A⁻¹B

Requires matrix multiplication and inverse

Recursive Equations

a_n = f(a_n-1)

Each term defined using previous terms

Differential Equations (Introduction)

dydx = f(x)

Involves derivatives (calculus level)

Word Problems to Equations

💡 Translation Guide

• "is" or "equals" → =

• "more than" → +

• "less than" → −

• "times" or "product" → ×

• "divided by" or "quotient" → ÷

• "of" (with percent) → ×

📋 Quick Reference Summary

Equation Type	General Form	Primary Method	Grade Level
Linear	ax + b = c	Inverse operations	6-9
Quadratic	ax² + bx + c = 0	Factoring, Quadratic Formula	9-11
Systems	Multiple equations	Substitution, Elimination	8-10
Polynomial	a_nxⁿ + ... = 0	Factoring, Synthetic Division	10-12
Rational	P(x)/Q(x) = R(x)/S(x)	Multiply by LCD	10-11
Radical	√f(x) = g(x)	Squaring (check solutions!)	10-11
Exponential	a^x = b	Logarithms	11-12
Logarithmic	log_ax = b	Convert to exponential	11-12
Absolute Value	\|f(x)\| = a	Split into cases	9-10

↑

🎯 Complete K-12 Algebraic Equations 🎯

🔍 Quick Search for Any Equation Type

📑 Equation Types & Topics

1. Linear Equations (Grade 6-9)

Standard Form

Solving Methods

One-Step Equations

Two-Step Equations

Example: Solve 3x + 7 = 22

Multi-Step Equations

Example: Solve 2(x − 3) + 5 = 4x − 1

Equations with Fractions

Example: Solve x3 + x4 = 14

Special Cases

Identity (Infinite Solutions)

Contradiction (No Solution)

2. Quadratic Equations (Grade 9-11)

Standard Form

Solving Methods

Method 1: Factoring

Example: Solve x² + 7x + 12 = 0

Method 2: Quadratic Formula

Example: Solve 2x² + 5x − 3 = 0

Method 3: Completing the Square

Example: Solve x² + 6x − 7 = 0

Method 4: Square Root Method

The Discriminant

Sum and Product of Roots

3. Systems of Equations (Grade 8-10)

Types of Systems

Linear Systems (2 Variables)

Method 1: Substitution

Example: Solve {x + 2y = 7 and 3x − y = 5}

Method 2: Elimination (Addition/Subtraction)

Example: Solve {2x + 3y = 12 and 4x − y = 10}

Method 3: Graphing

Types of Solutions

One Solution

No Solution

Infinite Solutions

3-Variable Systems

Non-Linear Systems

4. Polynomial Equations (Grade 10-12)

General Form

Cubic Equations (Degree 3)

Factoring Method

Example: Solve x³ − 6x² + 11x − 6 = 0

Quartic Equations (Degree 4)

Special Type: Biquadratic

Example: Solve x⁴ − 5x² + 4 = 0

Important Theorems

Remainder Theorem

Factor Theorem

Rational Root Theorem

Synthetic Division

Sum and Product of Roots

5. Rational Equations (Grade 10-11)

Definition

Solving Method

Example: Solve 3x−2 + 1x+1 = 2x²−x−2

Cross Multiplication

Example: Solve 2x+1x−3 = 52

Work Rate Problems

Extraneous Solutions

6. Radical Equations (Grade 10-11)

Definition

Solving Square Root Equations

Example: Solve 2x + 3 = 5

Equations with Two Radicals

Example: Solve x + 7 = x − 5

Equations Requiring Multiple Squarings

Cube Root Equations

Example: Solve ∛(2x − 1) = 3

Power Equations

💡 Important Tips

7. Exponential & Logarithmic Equations (Grade 11-12)

Exponential Equations

Method 1: Same Base

Example: Solve 2x+1 = 8

Method 2: Using Logarithms

Example: Solve 2^x+1 = 8

Example: Solve 3^x = 20