Subject
Algebra II
The quadratic formula, rational and radical expressions, logarithms, and complex numbers.
Begin at I if you’re new to Algebra II.
The Quadratic Formula
Solve any quadratic equation using x = (-b ± √(b² − 4ac)) / (2a). Always works, even when factoring fails.
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Completing the Square
Rewrite x² + bx + c as (x + h)² + k. Take half the middle coefficient, square it, balance the constant.
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The Discriminant
Compute Δ = b² − 4ac. Tells you how many real solutions a quadratic has, without solving it.
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Function Composition
Apply one function to the output of another: f(g(x)) means do g first, then f.
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Inverse Functions
Find f⁻¹(x) by swapping x and y, then solving for y. The inverse undoes what the original function did.
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Polynomial Long Division
Divide polynomials term by term: divide, multiply, subtract, bring down. Just like numeric long division.
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Synthetic Division
Fast shortcut for dividing by (x − c). Uses just the coefficients — bring down, multiply, add.
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Simplifying Rational Expressions
Factor numerator and denominator completely, then cancel any common factors. Cancel factors, never terms.
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Multiplying and Dividing Rational Expressions
Factor first, cancel common factors across both fractions, then multiply. For division, flip the second fraction first.
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Adding and Subtracting Rational Expressions
Find a common denominator, rewrite each fraction, then combine numerators. Same idea as numeric fractions, with polynomials.
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Rational Exponents
x^(m/n) means the nth root of x to the m power. Take the root first, then raise to the power.
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Imaginary and Complex Numbers
i = √(-1). Add and subtract by parts, multiply with FOIL, and replace i² with -1. Powers of i cycle every 4.
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Logarithm Properties and Evaluating
log_b(x) = y means b^y = x. Evaluate logs by asking 'b to what power gives x?' Use product, quotient, and power rules to simplify.
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Solving Exponential and Logarithmic Equations
Match bases or convert to exponential form. b^x = N ↔ rewrite N as b^k. log_b(x) = y ↔ x = b^y.
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