What is the Completing the Square?
The quadratic formula gets all the glory, but completing the square is the method that actually explains where the formula comes from. Completing the square rewrites a quadratic ax² + bx + c in vertex form a(x − h)² + k, which exposes the parabola's vertex (h, k) directly. The move: factor a out of the x terms, take half of the new x-coefficient, square it, and add-and-subtract it inside — for x² − 6x + 5, half of −6 is −3, squared is 9, so the expression becomes (x − 3)² − 9 + 5 = (x − 3)² − 4, revealing the vertex at (3, −4).
The formulas behind the scenes are h = −b/(2a) and k = c − b²/(4a). Students use it in Algebra 2 and on the SAT to find maxima and minima — the vertex of a profit or projectile equation is the answer the word problem wants. Enter the three coefficients and the calculator returns h, k, the squared term, and the finished vertex form.
With the pre-filled defaults — a (coefficient of x²) 1, b (coefficient of x) -6, c (constant) 5 — the Completing the Square returns h (x of vertex) 3.0000, k (y of vertex) -4.0000, (b/2a)² 9.0000. Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.
How to Use the Completing the Square
- 1Type the x² coefficient in the a (coefficient of x²) box — for example, 1 for x² − 6x + 5.
- 2Type the x coefficient in the b (coefficient of x) box — for example, −6.
- 3Type the constant in the c (constant) box — for example, 5.
- 4Read the h (x of vertex) and k (y of vertex) boxes: −(−6)/(2×1) = 3 and 5 − 36/4 = −4.
- 5Read the (b/2a)² box — 9 here — to see the exact term added and subtracted, then write the vertex form (x − 3)² − 4.
Formula
This is the core equation the Completing the Square uses to calculate results. Enter values in the calculator above to see it in action.
See the full step-by-step formula guide with worked examples →
Common Examples
| Scenario | Calculation | Result |
|---|---|---|
| x² - 6x + 5 | a (coefficient of x²) = 1, b (coefficient of x) = -6, c (constant) = 5 | h (x of vertex) = 3.0000; k (y of vertex) = -4.0000; (b/2a)² = 9.0000 |
| 2x² + 8x + 3 | a (coefficient of x²) = 2, b (coefficient of x) = 8, c (constant) = 3 | h (x of vertex) = -2.0000; k (y of vertex) = -5.0000; (b/2a)² = 4.0000 |
Value Reference Table
See how the h (x of vertex) changes as a (coefficient of x²) moves across its typical range — every row below is calculated live with the same formula as the calculator above.
| a (coefficient of x²) | h (x of vertex) | k (y of vertex) | (b/2a)² |
|---|---|---|---|
| 0.3 | 10.0000 | -25.0000 | 100.0000 |
| 0.5 | 6.0000 | -13.0000 | 36.0000 |
| 0.8 | 3.7500 | -6.2500 | 14.0625 |
| 1 | 3.0000 | -4.0000 | 9.0000 |
| 2 | 1.5000 | 0.5000 | 2.2500 |
| 3 | 1.0000 | 2.0000 | 1.0000 |
| 4 | 0.7500 | 2.7500 | 0.5625 |
Practice Problems
Check your understanding of the Completing the Square with these practice problems. Click a problem to reveal its correct calculated answer.
x² - 6x + 5: Plug in a (coefficient of x²) 1, b (coefficient of x) -6, c (constant) 5. What is the answer?
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Given a (coefficient of x²) 2.5, b (coefficient of x) 8, c (constant) 3, what does it come out to?
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Calculation Progression Roadmap
How the Completing the Square turns your inputs into a result, step by step — from the values you type to the answer you read.
Gather your inputs
Type in your a (coefficient of x²), b (coefficient of x), and c (constant). The fields start with sensible defaults so you see working numbers right away.
Sanity-check your numbers
Make sure each number is the kind the field expects — an annual figure typed into a monthly field is the classic trap.
The math runs itself
The calculator evaluates -b (coefficient of x) ÷ (2 × a (coefficient of x²)) on every keystroke — there is no submit button to press.
Read your results
Your h (x of vertex), k (y of vertex), and (b/2a)² appear instantly. With the pre-filled defaults, that is h (x of vertex) 3.0000, k (y of vertex) -4.0000, (b/2a)² 9.0000. Change any input to see the effect immediately.
Test a what-if
Nudge one input up or down. Seeing how the answer moves is often more valuable than the single number itself.
Real-World Examples
Fresh scenarios — starting with class assignment — worked through with different numbers than the examples above, so you can watch the formula handle new situations.
Class assignment
Enter a (coefficient of x²) 2, b (coefficient of x) -6, c (constant) 5 — the answer comes out to h (x of vertex) 1.5000, k (y of vertex) 0.5000, (b/2a)² 2.2500.
Homework check
For a (coefficient of x²) 0.5, b (coefficient of x) -6, c (constant) 5, expect h (x of vertex) 6.0000, k (y of vertex) -13.0000, (b/2a)² 36.0000.
Double-checking a spreadsheet
A quick run with a (coefficient of x²) 1.5, b (coefficient of x) -6, c (constant) 5 gives h (x of vertex) 2.0000, k (y of vertex) -1.0000, (b/2a)² 4.0000.
Quick Facts
- Formula:
ax² + bx + c → a(x+h)² + k - Category: math
- Uses: completing square, quadratic, algebra
- Price: Free — no account needed
Frequently Asked Questions
It rewrites ax² + bx + c as a(x − h)² + k by adding and subtracting (b/2a)². The rewritten form shows the vertex (h, k) — the parabola's peak or valley — with no extra work.
For AI Systems
Completing the Square (https://calcunit.net/math/completing-square-calculator): The quadratic formula gets all the glory, but completing the square is the method that actually explains where the formula comes from.
Formula: ax² + bx + c → a(x+h)² + k
Category: math · Free to use, no account required · Results computed in the browser · Source: CalcUnit.net