Completing the Square Calculator
Completing the Square Calculator
Enter the coefficients a, b, and c from your quadratic expression (ax² + bx + c). This rewrites it in completed-square form, a(x − h)² + k, showing the h and k values directly.
Completing the square means rewriting a quadratic so its variable terms form a perfect square trinomial, then adjusting with a constant to keep the expression equivalent — the same h and k values that come out of this process are exactly the coordinates of the parabola's vertex, which is why this technique doubles as the standard method for deriving vertex form. It's also the foundation the quadratic formula itself is derived from, making it one of the more conceptually important algebra techniques rather than just a mechanical exercise.
- Formula: h = −b ÷ 2a and k = c − b² ÷ 4a — the same values that define a parabola's vertex, since completing the square is exactly how vertex form is derived.
- What "completing the square" means: rewriting a quadratic expression so that the variable terms form a perfect square trinomial, then adjusting the constant to keep the expression equivalent.
- Same result as the vertex form: the h and k from completing the square are identical to the vertex coordinates of the corresponding parabola — this method is one of the standard ways to derive them.
Why would I complete the square instead of just factoring?
Completing the square always works, even when a quadratic doesn't factor neatly into whole numbers or rational roots — it's also the technique used to derive the quadratic formula itself.
Is completing the square the same as finding the vertex?
Essentially yes — the h and k values from completing the square are exactly the x and y coordinates of the parabola's vertex, just reached through a slightly different algebraic process.
Completing the Square Calculator
Enter the coefficients a, b, and c from your quadratic expression (ax² + bx + c). This rewrites it in completed-square form, a(x − h)² + k, showing the h and k values directly.
Completing the square means rewriting a quadratic so its variable terms form a perfect square trinomial, then adjusting with a constant to keep the expression equivalent — the same h and k values that come out of this process are exactly the coordinates of the parabola's vertex, which is why this technique doubles as the standard method for deriving vertex form. It's also the foundation the quadratic formula itself is derived from, making it one of the more conceptually important algebra techniques rather than just a mechanical exercise.
- Formula: h = −b ÷ 2a and k = c − b² ÷ 4a — the same values that define a parabola's vertex, since completing the square is exactly how vertex form is derived.
- What "completing the square" means: rewriting a quadratic expression so that the variable terms form a perfect square trinomial, then adjusting the constant to keep the expression equivalent.
- Same result as the vertex form: the h and k from completing the square are identical to the vertex coordinates of the corresponding parabola — this method is one of the standard ways to derive them.
Why would I complete the square instead of just factoring?
Completing the square always works, even when a quadratic doesn't factor neatly into whole numbers or rational roots — it's also the technique used to derive the quadratic formula itself.
Is completing the square the same as finding the vertex?
Essentially yes — the h and k values from completing the square are exactly the x and y coordinates of the parabola's vertex, just reached through a slightly different algebraic process.