Nonlinear Algebra

Five topics on nonlinear algebra, the SAT's Advanced Math domain and the densest content on the test. Exponents and radicals, expanding and factoring polynomials, quadratics and the parabola, rational, radical, and absolute-value equations, and nonlinear functions. This is where the freshman's dream lives, the belief that a power or a root spreads across a sum, and where a difference of squares quietly splits into two conjugate factors.

Topics

2.1

Exponents and Radicals

Lesson

Diagnostic

after diagnostic

2.2

Polynomial Expressions and Factoring

Lesson

Diagnostic

after diagnostic

2.3

Quadratic Equations and the Parabola

Rational, Radical, and Absolute-Value Equations

Nonlinear Functions and Function Notation

Lesson

Diagnostic

after diagnostic

Reference The exponent, radical, factoring, quadratic, rational, and absolute-value rules behind the problems in this unit

Multiply like bases (add)

$x^{a}\cdot x^{b} = x^{a+b}$

Divide like bases (subtract)

$\dfrac{x^{a}}{x^{b}} = x^{a-b}$

Power of a power (multiply)

$(x^{a})^{b} = x^{ab}$

Power over a product, never a sum

$(xy)^{a} = x^{a}y^{a}$

Negative exponent (reciprocal)

$x^{-a} = \dfrac{1}{x^{a}}$

Fractional exponent (root)

$x^{\tfrac{m}{n}} = \sqrt[n]{x^{m}}$

Square of a sum (keep the middle term)

$(x+a)^{2} = x^{2}+2ax+a^{2}$

Difference of squares (conjugates)

$x^{2}-a^{2} = (x+a)(x-a)$

Factor a quadratic (multiply to the constant, add to the middle)

$x^{2}+(p+q)x+pq = (x+p)(x+q)$

Quadratic formula (keep the two branches and the $2a$)

$x = \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Discriminant (two real roots if positive, one repeated real root if zero, no real roots if negative)

$b^{2}-4ac$

Vertex form (vertex at $(h,k)$, sign inside flips)

$y = a(x-h)^{2}+k$

Axis of symmetry (the vertex x-coordinate)

$x = -\dfrac{b}{2a}$

Radical equation (square, then check every candidate in the original)

$\sqrt{f(x)} = g(x) \Rightarrow f(x) = g(x)^{2}$

Rational equation (exclude any value that makes a denominator zero)

$\dfrac{p(x)}{x-a}, \quad x \neq a$

Absolute value (two cases if the right side is positive, none if negative)

$|f(x)| = k \Rightarrow f(x) = \pm k \ \ (k > 0)$

Unit 2 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Mixed items pulled from across the unit's topics, with no labels showing which is which. It surfaces the misconceptions that still bite when you cannot tell what a question is testing.