IBEW Algebra & Functions Complete Guide

Linear systems, quadratic factoring, rational equations, and functions

The NJATC / IBEW Electrical Aptitude Test allocates 33 questions (46 minutes) strictly to Algebra and Functions. Pacing averages 83 seconds per question. Mastery of systematic equation manipulation, quadratic factoring, and function evaluation is required to achieve a qualifying stanine score of 4 or higher.

On the Test

Official NJATC test section consists of 33 algebra questions. Calculators are strictly prohibited. Every operation must be computed on scratch paper with zero algebraic sign errors.

•Perform identical operations to both sides of the equals sign to isolate the target variable.
•When clearing denominators in rational equations, multiply every term by the least common denominator (LCD).
•When multiplying or dividing an inequality by a negative number, always reverse the inequality symbol.
•In function notation f(x), replace every occurrence of the variable x with the input argument inside the parentheses.
•Factoring quadratics: for x^2 + bx + c = 0, find two factors of c that sum to b.
⚠Common trap: Sign errors during distribution: for example, distributing -3(2x - 5) produces -6x + 15, not -6x - 15. Forgetting to flip the sign on the constant term is the #1 cause of lost points.

Multi-Step Linear Equations & Isolating Variables

Linear equations require isolating the variable on one side. Always simplify each side first: distribute parentheses, combine like terms, and then use inverse operations.

Consider the equation: 4(2x - 3) + 7 = 3(x + 5). First distribute coefficients: 8x - 12 + 7 = 3x + 15. Combine constants on the left: 8x - 5 = 3x + 15.

Subtract 3x from both sides: 8x - 3x - 5 = 15, yielding 5x - 5 = 15. Add 5 to both sides: 5x = 20. Divide both sides by 5: x = 4. Always substitute the result back into the original expression to confirm equality.

Systems of Two Linear Equations

The aptitude test frequently tests systems of two linear equations in two variables. Two methods are used: substitution and elimination (linear combination).

Elimination is faster for standardized testing when equations are in standard form (Ax + By = C). Multiply one or both equations by constants so that the coefficients of one variable are opposites, then add the equations together.

Example: 2x + 3y = 13 and x - y = 4. Multiply the second equation by 3 to eliminate y: 3(x - y) = 3(4) -> 3x - 3y = 12. Add the two equations: (2x + 3y) + (3x - 3y) = 13 + 12 -> 5x = 25 -> x = 5. Substitute x = 5 into the second equation: 5 - y = 4 -> y = 1. The solution is (5, 1).

Polynomial Expansion (FOIL) and Factoring Quadratics

Multiplying binomials uses FOIL: First, Outer, Inner, Last. For (2x + 3)(x - 4): First is 2x*x = 2x^2; Outer is 2x*(-4) = -8x; Inner is 3*x = 3x; Last is 3*(-4) = -12. Combining like terms: 2x^2 - 5x - 12.

Factoring reverses this process. For standard quadratics x^2 + bx + c, identify two numbers that multiply to c and add to b. For x^2 - 5x - 14 = 0: factors of -14 that add to -5 are -7 and +2. Factored form: (x - 7)(x + 2) = 0, giving roots x = 7 and x = -2.

Difference of Two Squares: Any expression in the form a^2 - b^2 factors into (a - b)(a + b). Example: 4x^2 - 49 = (2x - 7)(2x + 7). Look for perfect squares immediately to save computation time.

Exponents, Radicals, and Powers of 10

Product Rule: x^a * x^b = x^(a + b). Quotient Rule: x^a / x^b = x^(a - b). Power of a Power: (x^a)^b = x^(a * b).

Negative Exponent Rule: x^(-n) = 1 / x^n. For example, 4^(-2) = 1 / 4^2 = 1 / 16. A negative exponent indicates reciprocal position, not a negative value.

Simplifying Radicals: Factor out the largest perfect square factor. Example: sqrt(72) = sqrt(36 * 2) = 6 * sqrt(2). When adding radicals, only like radicands can be combined: 3*sqrt(8) + 2*sqrt(18) = 3*(2*sqrt(2)) + 2*(3*sqrt(2)) = 6*sqrt(2) + 6*sqrt(2) = 12*sqrt(2).

Function Notation f(x) and Linear Slopes

Function notation f(x) represents the output of a relation given input x. If f(x) = 2x^2 - 3x + 4, finding f(-3) means substituting (-3) into every occurrence of x: f(-3) = 2(-3)^2 - 3(-3) + 4 = 2(9) + 9 + 4 = 18 + 13 = 31. Watch negative signs carefully when squaring inputs: (-3)^2 = +9.

Slope-Intercept Form: y = mx + b, where m is slope and b is the y-intercept (0, b). Slope is calculated as m = (y2 - y1) / (x2 - x1).

Parallel lines have identical slopes (m1 = m2). Perpendicular lines have negative reciprocal slopes (m1 * m2 = -1, or m2 = -1/m1). If a conduit line has slope 3/4, a perpendicular junction has slope -4/3.

Key Takeaways

✓Write out intermediate steps on scratch paper. Do not compute sign reversals mentally.
✓Check answers by plugging candidate choices back into the original equation.
✓Memorize perfect squares up to 15^2 (225) to factor quadratics instantly without trial and error.
✓When simplifying rational equations, multiply all terms by the LCD to eliminate all fractions in step 1.
✓Negative times negative is positive: double check all parentheses distribution signs.

Key Terms

Linear Equation

An algebraic equation of first degree whose graph is a straight line.

System of Equations

Two or more equations with shared variables solved simultaneously.

FOIL Method

Binomial multiplication order: First, Outer, Inner, Last terms.

Roots / Zeroes

The x-values that make a polynomial equal to zero.

Negative Exponent

x^(-n) = 1 / x^n; specifies the reciprocal of the positive power.

Slope (m)

Rate of vertical change over horizontal change: (y2 - y1) / (x2 - x1).

Negative Reciprocal

A number flipped and multiplied by -1; defines perpendicular slopes.

Stanine Score

Standard Nine scale (1–9) used by the Electrical Training ALLIANCE.

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