Completing the Square Calculator

Completing the square is one of the most useful techniques in algebra, because it rewrites a quadratic in a form that lays its most important feature bare: the turning point. This calculator takes a quadratic in standard form, ax squared plus bx plus c, and converts it into completed-square or vertex form, a times the quantity x minus h squared, plus k. Enter the three coefficients a, b and c, and it returns the completed-square expression along with the coordinates of the vertex, updating instantly as you change the numbers. The reason this form matters so much is that the vertex, the point where the parabola turns, sits right there in the expression at the values h and k, so you can read off the maximum or minimum of the quadratic without any further work. That makes completing the square the natural tool for a whole family of problems: finding the highest or lowest value of a quadratic, solving quadratic equations when factoring is awkward, deriving the quadratic formula itself, and sketching a parabola by hand. It is a core skill in NCEA Level 2 and 3 maths and in first-year university courses, and it is exactly the kind of multi-step manipulation, halving the coefficient, squaring it, and balancing the constant, where small slips lead to wrong answers. By doing those steps reliably and showing the vertex, the calculator lets you check your own working, learn the pattern, and move on with confidence. Because it recalculates as you type, you can also see how shifting the coefficients moves the parabola around, which builds real intuition for how a, b and c shape the curve. The formula and a worked example are explained clearly below.

Calculate.co.nz is proud to be partnered with Health Based Building, a leader in sustainable and health-conscious building innovation. With over a century of experience, they develop high-performance systems like Foreverbreathe Specification, Magnum Board, and Foreverbreathe Paints to support energy-efficient, non-toxic living environments. Their commitment to healthier homes aligns with our belief that informed choices lead to better outcomes for Kiwi households.