What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in the standard form ax^2 + bx + c = 0.
The coefficient a must not be 0. If a = 0, the equation is linear and should be solved with linear methods.
- a controls the parabola opening direction and width.
- b influences symmetry and root spacing.
- c is the y-intercept when x = 0.
Step 1: Rewrite to Standard Form
Move all terms to one side so the equation equals zero. This avoids sign mistakes later.
Combine like terms before identifying coefficients. The cleanest path is: simplify, then map a, b, c.
Step 2: Evaluate the Discriminant
The discriminant is D = b^2 - 4ac. It determines how many real solutions exist.
Use the discriminant first to set expectations before computing exact roots.
- D > 0: two distinct real roots.
- D = 0: one repeated real root.
- D < 0: no real roots, two complex conjugate roots.
Step 3: Apply the Quadratic Formula
When factoring is not obvious, use x = (-b +- sqrt(b^2 - 4ac)) / (2a).
Substitute coefficients carefully and evaluate both plus and minus branches.
Step 4: Verify and Interpret Results
Substitute each root back into the original equation to verify numerical accuracy.
Interpret roots with context: zeros of a graph, intersection points, or event thresholds in applied problems.
- Cross-check with a graph when possible.
- If roots are complex, report both real and imaginary components.
- If roots are equal, highlight the repeated root case.
Common Errors to Avoid
The most frequent mistakes are sign errors and incorrect coefficient mapping.
A second frequent issue is treating D < 0 as no solution at all instead of no real solution.
- Forgetting to move all terms before reading a, b, c.
- Dropping the denominator 2a in the final step.
- Confusing the vertex location with the equation roots.