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"One Solution (Quadratic System)" Explained

Discriminant (b^2-4ac) is equal to zero.

Key Idea: One Solution (Quadratic System)

A quadratic system has one real solution when the quadratic and the line (or another curve) touch at exactly one point — typically where the line is tangent to the parabola.

General Rule:

  • Use the discriminant:

Δ=b^2−4ac

  • If Δ=0, there is exactly one real solution.

Why this matters for the SAT:

The discriminant gives a quick way to determine how many solutions a quadratic system has without solving. A discriminant of zero indicates one point of intersection, often signaling a tangent line.

"Two Solution (Quadratic System)" Explained

Discriminant (b^2-4ac) is more than zero (a positive number).

Key Idea: Two Solutions (Quadratic System)

A quadratic system or equation has two real solutions when the discriminant is positive.

General Rule:

  • Discriminant formula:

    D=b^2−4ac

  • If D>0, there are two distinct real solutions.

  • On a graph, this means the parabola intersects the line (or x-axis) at two points.

Why this matters for the SAT:

Understanding the discriminant helps you predict the number of solutions without solving completely.
It’s a fast way to analyze quadratic equations and systems efficiently.

"No Solution (Quadratic System)" Explained

Discriminant (b^2-4ac) is less than zero (a negative number).

Key Idea: No Solution (Quadratic System)

A quadratic system has no real solution when the equation has no points of intersection.

General Rule:

  • Use the discriminant:

Δ=b^2−4ac

  • If Δ<0 (negative), the quadratic has no real solutions.

Why this matters for the SAT:

Discriminants help quickly determine the number of solutions without solving the equation fully. This is essential for analyzing quadratic graphs and systems efficiently.

"Quadratic System of Equations" Explained

A system of equations with a quadratic equation in it.

Key Idea: Quadratic System of Equations

A quadratic system of equations includes at least one quadratic equation (an equation with an x^2 term) and another equation, which may be linear or quadratic.

General Rule:

  • Solving a quadratic system means finding the points of intersection between the graphs of the equations.

  • Common methods include substitution or graphing.

  • The system can have 0, 1, or 2 solutions, depending on how the graphs intersect.

Why this matters for the SAT:

Quadratic systems test your ability to combine algebra and graphing skills.
Recognizing a system with a quadratic helps you choose the right solving strategy—often substitution—to find where the curves meet.