Solving Quadratic Equations by Completing the Square

Enter the coefficients a, b, and c from ax² + bx + c = 0 to solve by completing the square — get both roots, the discriminant, and the parabola's vertex.

Quick Facts

Core identity
x² + bx = (x + b/2)² − (b/2)²
Adding and subtracting (b/2a)² turns ax² + bx + c into a perfect-square trinomial plus a constant.
Discriminant
D = b² − 4ac
D > 0 gives two real roots, D = 0 gives one repeated root, D < 0 gives complex roots.

Your Results

Calculated
Root x₁
-
First solution
Root x₂
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Second solution
Discriminant
-
b² − 4ac
Vertex (h, k)
-
Turning point of the parabola

Ready

Enter a, b, and c, then press Calculate.

How the Solving Quadratic Equations by Completing the Square works

This calculator applies a mathematical formula or algorithm to transform your inputs into a derived result. Understanding the underlying method helps you verify outputs, spot input errors, and interpret results correctly.

Formula and method

Calculate solving quadratic equations by completing the square — enter your values and get an accurate result with the underlying formula.

Common sources of error

  • Unit mismatch: ensure all inputs use the same unit system (metric or imperial) throughout
  • Order of operations: when entering expressions, follow standard PEMDAS/BODMAS rules
  • Rounding early: avoid rounding intermediate results — carry full precision through to the final step

Checking your result

For any calculation, apply a quick reasonableness check: is the result the right order of magnitude? Does it have the right sign? Does it change in the expected direction when you increase an input? If any of these fail, recheck the inputs and formula interpretation.

Applications

Mathematical results rarely stand alone — they feed into larger calculations, models, or decisions. Label your output with its units and document the inputs used alongside it so you can reproduce or share the result reliably.

Frequently Asked Questions

How accurate are the results?
The Solving Quadratic Equations by Completing the Square applies a standard formula to your inputs — accuracy depends on how precisely you measure those inputs. For planning and estimation, results are reliable. For high-stakes or professional decisions, cross-check the output with a domain expert or primary source.

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