SAT: Nonlinear Systems Cheat Sheet

The core ideas of SAT: Nonlinear Systems distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Substitution into Nonlinear Equations

Replace one variable using the linear equation and substitute into the nonlinear equation to produce a single equation in one variable, typically a quadratic.

Discriminant

For $ax^2 + bx + c = 0$, the discriminant $D = b^2 - 4ac$ determines the number of real solutions: $D > 0$ gives two, $D = 0$ gives one, $D < 0$ gives none.

Line-Parabola Intersections

Setting a linear equation equal to a quadratic produces a quadratic equation whose solutions are the $x$-coordinates of intersection points. A tangent line gives $D = 0$.

Vieta's Formulas

For $ax^2 + bx + c = 0$ with roots $r_1$ and $r_2$: sum $= -\frac{b}{a}$ and product $= \frac{c}{a}$. These allow answering SAT questions without fully solving.

Circle-Line Systems

Substituting a linear equation into $x^2 + y^2 = r^2$ yields a quadratic in one variable. The discriminant determines 0, 1, or 2 intersection points.

Tangent Line to a Parabola

A line is tangent to a parabola when the resulting quadratic has exactly one solution, meaning $D = 0$. Solving $D = 0$ for the parameter finds the tangency condition.

Number of Solutions

The SAT often asks 'how many solutions' a system has. After substitution, use the discriminant: $D > 0$ yields 2, $D = 0$ yields 1, $D < 0$ yields 0.

Vertex Form and Horizontal Lines

A parabola $y = (x - h)^2 + k$ has its minimum (or maximum) at $y = k$. A horizontal line $y = c$ intersects at one point when $c = k$ and at two points when $c > k$ (for upward-opening).

Systems with Rational Functions

Setting $y = \frac{1}{x}$ equal to a linear equation $y = mx + b$ produces $\frac{1}{x} = mx + b$, which simplifies to $mx^2 + bx - 1 = 0$, a quadratic in $x$.

Absolute Value Intersections

The graph of $y = |x|$ is a V-shape. A horizontal line $y = c$ ($c > 0$) intersects it at two points ($x = \pm c$); $y = 0$ touches at one point; $y < 0$ gives no intersection.

Key Terms at a Glance

Nonlinear System:A system of equations where at least one equation is not linear (e.g., contains $x^2$, $\frac{1}{x}$, or $|x|$).

Discriminant:The expression $b^2 - 4ac$ from the quadratic formula, which determines the number and type of roots.

Substitution:A method for solving systems by expressing one variable in terms of the other and substituting into the second equation.

Tangent Line:A line that touches a curve at exactly one point without crossing it at that location.

Vieta's Formulas:Relationships between the coefficients and roots of a polynomial: for $ax^2 + bx + c = 0$, sum $= -b/a$ and product $= c/a$.

Vertex Form:The form $y = a(x - h)^2 + k$ of a quadratic, where $(h, k)$ is the vertex.

Intersection Point:A point that lies on both curves in a system, found by solving the system simultaneously.

Quadratic Equation:An equation of the form $ax^2 + bx + c = 0$ where $a \neq 0$.

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