Quadratic Equations Practice Problems & Factoring / Formula Solutions
A quadratic equation is a second-order polynomial equation of the form ax² + bx + c = 0. Master finding the roots (solutions) using factoring, completing the square, and the universal Quadratic Formula with discriminant proofs.
🧠 Core Problem-Solving Strategies
🧩 Sum and Product Factoring (Vieta)
For x² + bx + c = 0, find two numbers that multiply to c and add to b. If (x − r₁)(x − r₂) = 0, then x = r₁ or x = r₂.
⚡ The Universal Quadratic Formula
x = (−b ± √(b² − 4ac)) / (2a). Use when expressions cannot be easily factored.
🔍 Discriminant Analysis (Δ = b² − 4ac)
If Δ > 0, there are 2 distinct real roots. If Δ = 0, there is 1 double root. If Δ < 0, there are 2 complex roots.
❓ Frequently Asked Questions
What are the roots or zeros of a quadratic equation?
The roots (or zeros) are the values of x that make the equation equal to zero, representing the x-intercepts of the parabola on a Cartesian graph.
Why do the signs flip when finding roots from factored form (x − 3)(x + 4) = 0?
By the Zero-Product Property, if AB = 0, then either A = 0 or B = 0. Setting x − 3 = 0 gives x = +3, and x + 4 = 0 gives x = −4.
Continue Learning & Practicing
Reinforce your understanding with printable worksheets, step-by-step guides, calculators, and speed games: