Quadratic Equations — SAT Math Explained
An equation of the form ax² + bx + c = 0, where a ≠ 0. The highest power of the variable is 2, which causes its graph to form a U-shaped curve called a parabola.
The Core Idea
Quadratic equations can have zero, one, or two real solutions. This is fundamentally different from linear equations (always one solution or no unique solution). The discriminant determines which case you're in.
Key Vocabulary
Involving a squared variable (from the Latin 'quadratus', meaning square)
ax² + bx + c = 0
The x-values where the equation equals zero — where the parabola crosses the x-axis
The U-shaped graph of a quadratic
The turning point of the parabola — its maximum or minimum point
b² - 4ac — determines the nature and number of solutions
Discriminant Interpretation
Two distinct real solutions (parabola crosses x-axis twice)
One repeated real solution (parabola just touches x-axis at vertex)
No real solutions (parabola doesn't cross x-axis)
Methods To Solve
Factoring: fastest when factors are obvious
Square Root Method: best when equation is in form (x - h)² = k
Completing the Square: powerful for deriving vertex form
Quadratic Formula: always works for any quadratic
Real World Application
Maximizing the area of a garden given a fixed perimeter, finding when a launched object hits the ground, optimizing revenue in business
How This Connects to Other Topics
Links directly to factoring, the quadratic formula, and parabola graphs
Practice: Quadratic Equations
5 SAT-style questions. Select your answer and get an instant explanation.
Which of the following is a quadratic equation?
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