How this instrument works
Completing the square rewrites ax²+bx+c into vertex form, a(x−p)²+q, which reveals the parabola's vertex — its lowest or highest point — sitting right at the coordinates (p, q). The vertex's x-coordinate, p, comes from −b divided by 2a; the vertex's y-coordinate, q, is whatever remains of the constant term once the squaring adjustment (b² divided by 4a) is subtracted back out.
This page walks through that transformation directly for any quadratic, without needing to manually add and subtract the completing term by hand. The companion Completing the Square Works page on this site instead checks the DISCRIMINANT of the same quadratic, testing whether it has real roots at all — a related but genuinely different question from finding the vertex form itself.
Vertex form is worth having on hand beyond just locating the vertex: it's also the fastest route to solving a quadratic by hand (isolate the squared term and take a square root of both sides) and to graphing a parabola quickly, since the vertex and the direction it opens are both visible directly in the rewritten form.
- Enter the quadratic's three coefficients into the a, b, and c fields.
- Read Vertex x-coordinate: where the parabola's turning point sits horizontally.
- Read Vertex y-coordinate: the value of the quadratic at that turning point.
- Write the completed-square form as a(x − vertex x)² + vertex y, and expand it back out to confirm it reproduces the original quadratic.
Worked example — completing the square on x²−6x+5
For x²−6x+5 (a=1, b=−6, c=5), the vertex's x-coordinate is −(−6)⁄2=3, and its y-coordinate is 5−(−6)²⁄4=5−9=−4, giving the completed-square form (x−3)²−4. Expanding that back out confirms it: (x−3)²−4=x²−6x+9−4=x²−6x+5, matching the original exactly.
For a non-monic quadratic, 2x²+8x+3 (a=2, b=8, c=3): the vertex sits at x=−8⁄4=−2, and y=3−64⁄8=−5, giving 2(x+2)²−5. Expanding confirms it: 2(x+2)²−5=2x²+8x+8−5=2x²+8x+3.
Questions
What is completing the square used for?
Rewriting a quadratic into vertex form, which directly reveals the parabola's turning point (its vertex) and provides a quick route to solving the quadratic by hand, without needing the full quadratic formula.
How do you find the vertex from the completed-square form?
Once written as a(x−p)²+q, the vertex sits exactly at the coordinates (p, q) — no further calculation needed, since the squared term is zero precisely when x equals p, leaving q as the value at that point.
Does completing the square work for every quadratic?
Yes — every quadratic ax²+bx+c can be rewritten into vertex form this way, regardless of whether its roots end up being real or complex numbers; the transformation itself never fails.
How is this different from the Completing the Square Works page on this site?
That page checks the discriminant to test whether the quadratic has real roots at all; this page performs the actual vertex-form transformation itself, a related but separate calculation.
Why is vertex form useful for graphing?
The vertex's coordinates are visible directly in the rewritten expression, and the leading coefficient's sign shows immediately whether the parabola opens upward or downward — both readable at a glance, without plotting any points first.