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Key Features
Instant Calculation
Get accurate results in real time with our optimized algorithm.
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Vertex form immediately reveals a parabola's highest or lowest point, which standard form hides behind three separate coefficients. This calculator converts a quadratic from standard form (ax² + bx + c) into vertex form (a(x − h)² + k), showing the vertex coordinates directly, with a full step-by-step solution.
How it works
Enter the standard form coefficients a, b, and c. The calculator completes the square algebraically, finding the vertex x-coordinate h = −b/2a and y-coordinate k = c − b²/4a.
- Enter a (coefficient of x²).
- Enter b (coefficient of x).
- Enter c (constant).
- Click Calculate to see your results.
Examples
Converting 2x² − 8x + 3
This converts to y = 2(x − 2)² − 5, with a vertex at (2, −5) — the parabola's minimum point since a is positive.
Who should use it
- Finding the maximum or minimum value of a quadratic function for an optimization problem.
- Converting between standard and vertex form for algebra or precalculus homework.
Industry applications
- Algebra and precalculus education
- Physics (projectile motion) and engineering optimization problems
Advantages
- Shows both the full vertex form equation and the individual vertex coordinates.
- Uses the exact algebraic completing-the-square formulas, not an approximation.
Limitations
- Only applies to quadratics (degree-2 polynomials) — cubic or higher-degree equations have no vertex form.
Common mistakes to avoid
- Forgetting the negative sign inside the formula for h, since h = −b/2a (not +b/2a).
- Mixing up h and k with the standard form coefficients b and c — they represent the vertex coordinates, not the original coefficients.
Best practices
- After converting, plug the vertex coordinates back into the original standard-form equation to double-check the y-value matches.
Tips
- The vertex's y-coordinate, k, directly gives the maximum or minimum value of the function — no need to evaluate the original equation separately.