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Algebra Essentials

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1 Introduction to the OAR

About the OAR Who Takes the OAR and Why Study Plan and Test-Day Strategy

2 Math Skills Test (MST)

Arithmetic Fundamentals Fractions, Ratios, and Proportions Percentages and Real-World Applications Algebra Essentials Geometry Foundations Word Problems, Statistics, and Data Interpretation

3 Reading Comprehension Test (RCT)

RCT Overview and Core Strategies Main Idea and Purpose Questions Detail, Fact, and Inference Questions Vocabulary in Context and Author Tone Advanced Reading Strategies and Passage Types RCT Practice Passages and Questions

4 Mechanical Comprehension Test (MCT)

Forces and Newton's Laws of Motion Energy, Work, and Power Simple Machines and Mechanical Advantage Gears, Belts, and Rotational Motion Fluids, Pressure, and Hydraulics Electricity, Circuits, and Magnetism Motion, Heat, and Thermodynamics MCT Practice Problems and Review

5 Full Practice Tests and Final Review

Practice Test: Math Skills Practice Test: Reading Comprehension Practice Test: Mechanical Comprehension Final Review and Test-Day Preparation

Algebra Essentials

Key Takeaways Explain This Topic AI

Algebra on the OAR ranges from basic single-variable equations to systems of equations and quadratic expressions.

The golden rule: whatever you do to one side of an equation, you must do to the other side.

FOIL (First, Outer, Inner, Last) is essential for expanding binomials — practice until it is automatic.

Factoring quadratics and recognizing special patterns (difference of squares, perfect square trinomials) saves significant time.

Word problems require translating English into algebraic expressions — read carefully and define variables before writing equations.

Last updated: March 2026

Algebra is the largest topic area on the OAR Math Skills Test. You will encounter everything from simple one-step equations to more complex multi-step problems involving quadratics and systems of equations.

  • Solving Single-Variable Equations
  • One-Step Equations
  • Equation Operation Solution
  • x + 7 = 15 Subtract 7 x = 8
  • x - 4 = 12 Add 4 x = 16
  • 3x = 27 Divide by 3 x = 9
  • x/5 = 8 Multiply by 5 x = 40
  • Two-Step Equations

Strategy: Undo addition/subtraction first, then multiplication/division.

  • Example: 4x + 7 = 31
  • Subtract 7: 4x = 24
  • Divide by 4: x = 6
  • Example: (x - 3)/5 = 4
  • Multiply by 5: x - 3 = 20
  • Add 3: x = 23
  • Multi-Step Equations
  • Example: 3(2x - 4) = 5x + 2
  • Distribute: 6x - 12 = 5x + 2
  • Subtract 5x: x - 12 = 2
  • Add 12: x = 14
  • Example: 2(x + 3) - 4(x - 1) = 10
  • Distribute: 2x + 6 - 4x + 4 = 10
  • Combine: -2x + 10 = 10
  • Subtract 10: -2x = 0
  • Divide: x = 0
  • Exponents and Roots
  • Exponent Rules
  • Rule Expression Result
  • Product rule x³ × x⁴ x⁷ (add exponents)
  • Quotient rule x⁶ / x² x⁴ (subtract exponents)
  • Power rule (x³)² x⁶ (multiply exponents)
  • Zero exponent x⁰ 1 (any nonzero base)
  • Negative exponent x⁻² 1/x²
  • Fractional exponent x^(1/2) √x
  • Common Powers to Memorize
  • Base Squared Cubed
  • 2 4 8
  • 3 9 27
  • 4 16 64
  • 5 25 125
  • 6 36 216
  • 7 49 343
  • 8 64 512
  • 9 81 729
  • 10 100 1,000
  • 11 121 1,331
  • 12 144 1,728
  • Square Roots
  • √ Value √ Value
  • √1 1 √49 7
  • √4 2 √64 8
  • √9 3 √81 9
  • √16 4 √100 10
  • √25 5 √121 11
  • √36 6 √144 12
  • Simplifying Radicals

Find the largest perfect square factor:

  • √72 = √(36 × 2) = 6√2
  • √50 = √(25 × 2) = 5√2
  • √48 = √(16 × 3) = 4√3
  • FOIL and Polynomial Multiplication
  • FOIL Method
  • (a + b)(c + d) = ac + ad + bc + bd
  • Example: (x + 3)(x + 5)
  • First: x × x = x²
  • Outer: x × 5 = 5x
  • Inner: 3 × x = 3x
  • Last: 3 × 5 = 15
  • Result: x² + 5x + 3x + 15 = x² + 8x + 15
  • Special Products
  • Pattern Expansion
  • (a + b)² a² + 2ab + b²
  • (a - b)² a² - 2ab + b²
  • (a + b)(a - b) a² - b² (difference of squares)
  • Example: (x + 4)² = x² + 8x + 16
  • Example: (x + 7)(x - 7) = x² - 49
  • Factoring
  • Factoring Trinomials

To factor x² + bx + c, find two numbers that multiply to c and add to b.

  • Example: x² + 7x + 12
  • Need: two numbers that multiply to 12 and add to 7
  • Answer: 3 and 4
  • Factored: (x + 3)(x + 4)
  • Example: x² - 5x + 6
  • Need: two numbers that multiply to 6 and add to -5
  • Answer: -2 and -3
  • Factored: (x - 2)(x - 3)
  • Factoring with Leading Coefficient

For ax² + bx + c where a ≠ 1, use the AC method:

  • Example: 2x² + 7x + 3
  • Multiply a × c = 2 × 3 = 6
  • Find factors of 6 that add to 7: 6 and 1
  • Rewrite: 2x² + 6x + x + 3
  • Factor by grouping: 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
  • Quadratic Equations
  • The Quadratic Formula

For ax² + bx + c = 0:

  • x = (-b ± √(b² - 4ac)) / 2a
  • Example: Solve 2x² - 5x - 3 = 0
  • a = 2, b = -5, c = -3
  • Discriminant: (-5)² - 4(2)(-3) = 25 + 24 = 49
  • x = (5 ± 7) / 4
  • x = 12/4 = 3 or x = -2/4 = -1/2
  • The Discriminant

b² - 4ac tells you the nature of the roots:

  • Discriminant Roots
  • Positive Two distinct real roots
  • Zero One repeated real root
  • Negative No real roots (complex)
  • Systems of Equations
  • Substitution Method
  • Solve one equation for one variable
  • Substitute into the other equation
  • Solve and back-substitute

Example:

  • y = 2x + 1
  • 3x + y = 16
  • Substitute: 3x + (2x + 1) = 16 → 5x + 1 = 16 → 5x = 15 → x = 3
  • Back-substitute: y = 2(3) + 1 = 7 Solution: (3, 7)
  • Elimination Method
  • Align equations
  • Multiply to make one variable's coefficients opposites
  • Add equations to eliminate that variable

Example:

  • 2x + 3y = 12
  • 4x - 3y = 6
  • Add directly (y cancels): 6x = 18 → x = 3
  • Substitute: 2(3) + 3y = 12 → 3y = 6 → y = 2 Solution: (3, 2)
  • Inequalities
  • Solving Linear Inequalities

The same as equations, with one critical exception: flip the inequality sign when multiplying or dividing by a negative number.

  • Example: -3x + 6 > 15
  • Subtract 6: -3x > 9
  • Divide by -3 (flip!): x < -3
  • Compound Inequalities
  • "And" (intersection): Both conditions must be true: -2 < x < 5
  • "Or" (union): At least one condition is true: x < -1 or x > 4