Quadratic equations can be solved by factoring, completing the square, or the quadratic formula. Choose the method based on structure, and use the discriminant b^2 - 4ac to predict the number and type of real solutions.
- Set equations to zero before solving.
- Factoring is fastest when integer factors are visible.
- The discriminant signals two, one, or zero real roots.
Select an efficient method
If the trinomial factors cleanly over integers, factoring is quickest. If not, completing the square or the quadratic formula gives a guaranteed path.
For equations already in vertex-like structure, completing the square can preserve interpretation and reduce arithmetic complexity.
Use the discriminant for root behavior
For ax^2 + bx + c = 0, discriminant D = b^2 - 4ac. If D > 0, two distinct real roots; if D = 0, one repeated real root; if D < 0, complex conjugate roots.
- Always identify a, b, c with signs included.
- Write exact roots first, then decimal approximations if needed.
- Check roots in the original equation.
Solve using the quadratic formula
Solve 2x^2 - 3x - 2 = 0 from a projectile height model snapshot.
- Identify a = 2, b = -3, c = -2.
- Compute discriminant: b^2 - 4ac = 9 + 16 = 25.
- Apply formula: x = (3 +/- 5)/4, so roots are x = 2 and x = -1/2.
- Substitute each root to verify the equation equals zero.
Common mistakes
- Forgetting to set one side to zero before factoring.
- Copying the sign of b incorrectly into the formula.
- Arithmetic errors in b^2 - 4ac.
- Dropping one root when using square roots or formula.
Try one
How many real solutions does x^2 + 6x + 9 = 0 have?
One repeated real solution, because discriminant is 36 - 36 = 0. The root is x = -3.
Sources
- OpenStax: College AlgebraOpen textbook with full chapter coverage, worked examples, and practice problems.
- Khan Academy: Algebra and Algebra 2Official lessons and practice sets for core algebra skills.