Domain 4 • Non-Linear Functions & Quadratics
Algebra 2 on ALEKS
Algebra 2 is the gateway to Pre-Calculus placement (scores 61+). Questions test non-linear relationships: quadratics, roots, rational equations, logarithms, and imaginary numbers.
Core Tested Concepts
The Quadratic Formula & Discriminantx = (-b ± √(b² - 4ac)) / (2a). Understanding that discriminant b² - 4ac determines whether roots are real, repeated, or complex.
Complex Numbers & Imaginary Unit ii = √(-1), i² = -1, i³ = -i, i⁴ = 1. Operations with complex numbers (a + bi) and multiplying by conjugates.
Logarithms & Exponential EquationsLog properties: log(xy) = log x + log y, log(x/y) = log x - log y, log(xⁿ) = n·log x. Solving 2^x = y.
Radicals & Rational Exponentsx^(m/n) = n√(xᵐ), simplifying radicals, solving equations with radical terms, checking for extraneous roots.
High-Yield Cheat Sheet
Essential Algebra 2 Formulas
Quadratic Formula
x = (-b±√(b²-4ac))/2a
Log Product Rule
log(ab) = log a + log b
Imaginary Unit
i² = -1, i⁴ = 1
Practice Drills
5 Authentic Algebra 2 Practice Questions
1Practice Problem
Solve the radical equation for x: √(4x + 9) - 1 = 6
Hint: First isolate the radical by adding 1 to both sides: √(4x + 9) = 7. Then square both sides.
Formula: √(f(x)) = c => f(x) = c²
2Practice Problem
Evaluate the exact logarithmic value: log₃(81) + log₄(1/16)
Hint: 3⁴ = 81 (so log₃ 81 = 4) and 4⁻² = 1/16 (so log₄ 1/16 = -2).
Formula: log_b(bⁿ) = n
3Practice Problem
Simplify the complex number division: (10 + 5i) / (1 + 2i)
Hint: Multiply numerator and denominator by the complex conjugate of the denominator: (1 - 2i).
Formula: (a+bi)(a-bi) = a² + b²
4Practice Problem
For what positive value of x is 2^(3x - 1) = 32?
Hint: Express 32 as a power of 2: 32 = 2⁵. Then set exponents equal: 3x - 1 = 5.
Formula: bⁿ = bᵐ => n = m
5Practice Problem
Find the positive root of the quadratic equation x² - 6x - 16 = 0.
Hint: Factor the quadratic: find two numbers that multiply to -16 and add to -6.
Formula: (x - 8)(x + 2) = 0