Completing the Square Calculator

Completing the square rewrites a quadratic so the variable appears only once, inside a perfect square. Take 2x² + 8x + 5. Factor out the 2, halve the 4 that is left in front of x, square it to get 4, and the expression becomes 2(x + 2)² - 3. The vertex is now readable at a glance: (-2, -3).

Three sister tools cover the neighboring questions. The Quadratic Formula Calculator jumps straight to the roots when you do not need the working, the Factoring Calculator is faster when the quadratic breaks into whole-number factors, and the Square Root Calculator simplifies the radical that appears at the last step. Use this page when you need the method: the vertex, the maximum or minimum, or a step-by-step solution you can copy into your homework.

How completing the square works

Every quadratic in standard form can be rewritten with a single identity:

The bracket (x + b/2a)² is the perfect square, and the leftover constant c - b²/4a is what remains after the square has absorbed as much of the expression as it can. Written as a(x - h)² + k, the parabola has its vertex at (h, k), where h = -b/2a and k = c - b²/4a.

The mechanics are always the same four moves: factor the leading coefficient out of the first two terms, halve the x-coefficient that is left and square it, add and subtract that number inside the bracket, then group the perfect square and tidy the constants. The number you add, (b/2a)², is the only piece of real arithmetic in the whole method.

Worked example: 2x² + 8x + 5

Follow the six steps the calculator shows for the default example.

1. Standard form: a = 2, b = 8, c = 5.
2. Factor out a: 2(x² + 4x) + 5, because 8 ÷ 2 = 4.
3. Halve and square: (4 ÷ 2)² = 2² = 4. This is the number you add.
4. Add and subtract inside: 2(x² + 4x + 4 − 4) + 5.
5. Group the square: 2(x + 2)² − 8 + 5, because the −4 leaving the bracket is multiplied by 2.
6. Combine constants: 2(x + 2)² − 3.

Check it by expanding: 2(x + 2)² − 3 = 2(x² + 4x + 4) − 3 = 2x² + 8x + 8 − 3 = 2x² + 8x + 5. The original is back, so the rewrite is exact.

Now solve 2x² + 8x + 5 = 0 from the completed form. Set 2(x + 2)² − 3 = 0, so (x + 2)² = 3/2. Take the square root of both sides: x + 2 = ±√(3/2) = ±√6 ÷ 2. Therefore x = (−4 ± √6) ÷ 2, which is about −0.775 and −3.225. The discriminant b² − 4ac = 64 − 40 = 24 agrees: √24 = 2√6, and the quadratic formula gives (−8 ± 2√6) ÷ 4, the same pair after cancelling the 2.

A second worked example with fractions: x² + 5x + 3

An odd b is where students usually stumble, because half of it is a fraction. Here a = 1, so there is nothing to factor out. Half of 5 is 5/2, and squaring gives 25/4. Add and subtract it: x² + 5x + 25/4 − 25/4 + 3. Group the square and combine the constants, using 3 = 12/4, so −25/4 + 12/4 = −13/4. The completed form is (x + 5/2)² − 13/4, giving the vertex (−5/2, −13/4), or (−2.5, −3.25) as decimals.

Solving: (x + 5/2)² = 13/4, so x + 5/2 = ±√13 ÷ 2 and x = (−5 ± √13) ÷ 2. Since √13 ≈ 3.6056, the roots are about −0.697 and −4.303. Note how the exact answer keeps the radical: rounding √13 too early is the fastest way to lose marks on an exam. The calculator on this page keeps every intermediate value as an exact fraction and only rounds in the final approximation line.

Completed square form for common quadratics

Each row was produced with the identity above. The third column is what the calculator prints, and the last column solves the corresponding equation from that form.

Two patterns are worth noticing. First, the number you add depends only on b and a, never on c, which is why x² + 6x + 5 and x² + 6x + 13 both take a 9. Second, when a is negative the vertex is a maximum instead of a minimum: −2(x − 2)² + 5 peaks at y = 5.

The number you add: half of b, squared

For a monic quadratic (a = 1) the whole method collapses to a single reflex: halve b, square it, and you have a perfect square trinomial. This table is worth memorizing for mental math.

The squared value is always positive, even when b is negative, because squaring kills the sign. The sign of b survives only inside the bracket: a negative b gives (x − something)².

Reading the completed form: what a and k tell you

Once a quadratic sits in the form a(x − h)² + k, the signs of a and k answer three questions instantly, with no graphing and no discriminant.

The rule behind the table: real roots exist only when a and k have opposite signs, because solving needs (x − h)² = −k/a, and a square can never be negative. That is the same information the discriminant carries, arrived at without computing b² − 4ac.

How to use this calculator

1. Rearrange first. Move every term to one side so the expression reads ax² + bx + c. For example 2x² + 5 = 7x becomes 2x² − 7x + 5.
2. Enter a, b and c with their signs. A missing term simply means that coefficient is 0. Decimals such as 0.5 are accepted and converted to exact fractions.
3. Read the headline. The blue card shows the completed square form, the vertex, the axis of symmetry, the minimum or maximum value, and the discriminant.
4. Follow the numbered steps. Each one names what is happening and shows the algebra, so you can copy the working rather than just the answer.
5. Check the solving panel. It isolates the bracket, takes the square root of both sides, and gives the exact roots in radical or a ± bi form plus decimal approximations.
6. Try the example buttons if you want to see how a negative a, a fractional vertex, or a complex-root case behaves before entering your own numbers.

Everything recalculates as you type, so you can watch how one coefficient changes the whole picture.

Who this calculator is for

• Algebra 1 and Algebra 2 students learning the method and needing every step spelled out.
• Precalculus and calculus students converting to vertex form before graphing, or completing the square inside an integral or a conic-section equation.
• Students checking homework who already have an answer and want to confirm the vertex and the constant they added.
• Teachers and tutors generating clean worked examples with exact fractions instead of rounded decimals.
• Anyone optimizing a quadratic model such as maximum area, maximum revenue, or projectile height, where the vertex is the answer.

Key terms explained

• Standard form: ax² + bx + c, with terms ordered by descending power and a not equal to 0.
• Vertex form: a(x − h)² + k. The completed square form is exactly this, with h = −b/2a and k = c − b²/4a.
• Perfect square trinomial: an expression such as x² + 6x + 9 that factors as a single square, (x + 3)². Completing the square manufactures one on purpose.
• Vertex: the turning point of the parabola, the minimum when a is positive and the maximum when a is negative.
• Axis of symmetry: the vertical line x = h that mirrors the parabola onto itself.
• Discriminant: b² − 4ac. Positive means two real roots, zero means one repeated root, negative means a complex conjugate pair.
• Monic quadratic: one with a = 1, where the method needs no factoring step at all.

What changes the result the most

• b relative to a: the whole method hinges on b/2a. Change b and both the number you add and the horizontal position of the vertex move.
• The sign of a: it flips the parabola, turning a minimum into a maximum and reversing which side of the x-axis produces real roots.
• c: it shifts the parabola vertically without touching h. Raise c enough and two real roots become none.
• Whether b/2a is a whole number: an odd b with a = 1 forces fractions, which is where most hand-worked errors appear.
• Whether −k/a is a perfect square: if it is, the roots are rational and the quadratic could also have been factored; if not, you are left with a radical.

Tips for doing it by hand

• Factor a out of only the x² and x terms. The constant c stays outside the bracket the entire time.
• Write the bracket as (x + b/2a)² immediately once you know b/2a. The sign inside the bracket matches the sign of b/2a.
• When the subtracted number leaves the bracket, multiply it by a. Forgetting that factor is the single most common error.
• Keep fractions as fractions. Converting 5/2 to 2.5 is fine, but 1/3 to 0.333 will poison the final constant.
• Always expand your answer back out as a check. It takes ten seconds and catches nearly every slip.
• If you only need the vertex and not the working, remember h = −b/2a and substitute it back to get k.

Deriving the quadratic formula by completing the square

The quadratic formula is not a separate idea, it is this method applied once to the general equation. Start with ax² + bx + c = 0 and divide through by a to get x² + (b/a)x + c/a = 0. Move the constant across: x² + (b/a)x = −c/a. Half of b/a is b/2a, so add (b/2a)² = b²/4a² to both sides. The left side is now the perfect square (x + b/2a)², and the right side is b²/4a² − c/a, which over the common denominator 4a² equals (b² − 4ac)/4a². Take the square root of both sides: x + b/2a = ±√(b² − 4ac) ÷ 2a. Subtract b/2a and you have the familiar x = (−b ± √(b² − 4ac)) ÷ 2a.

That is why the discriminant appears under the radical and why the axis of symmetry is −b/2a: both fall out of the same completion. If you would rather skip the derivation and go straight to the roots, the Quadratic Formula Calculator does exactly that.

Where completing the square is actually used

Beyond homework, the technique earns its keep in several places. In optimization problems the vertex is the answer: if revenue is modeled by −2x² + 8x − 3, the completed form −2(x − 2)² + 5 says the maximum revenue of 5 occurs at x = 2, with no calculus required. In graphing, vertex form gives the turning point, the axis of symmetry, and the direction of opening before you plot a single point. In analytic geometry, completing the square on both variables turns a scattered equation such as x² + y² − 6x + 4y = 3 into the standard circle form (x − 3)² + (y + 2)² = 16, revealing the center and radius. In calculus, it rewrites awkward denominators so a substitution becomes obvious. And in statistics, completing the square is the standard algebraic step in deriving the normal distribution and in least-squares regression. The same four moves keep reappearing.

Limitations and assumptions

• The tool handles a single-variable quadratic only. It does not complete the square in two variables, so circle and ellipse equations must be done one variable at a time.
• a cannot be 0. With no x² term there is no square to complete, and the expression is linear instead.
• Decimal coefficients are converted to fractions with up to six decimal places. Coefficients with longer expansions are rounded before the algebra runs, and the result panel says so.
• Radicals are simplified for reasonably sized integers. Extremely large values under the radical may be left unsimplified rather than fully factored.
• Complex roots are shown in a ± bi form, but the tool does no further complex arithmetic.
• The calculator rewrites and solves, it does not draw the parabola. Use the vertex, the axis of symmetry, and the y-intercept it reports to sketch one yourself.

How it compares to related calculators

This page answers "rewrite this quadratic in vertex form and show me every step". If your question is different, a sister tool fits better:

• For the roots alone, with the discriminant and no completion steps, use the Quadratic Formula Calculator.
• When the quadratic factors over the integers, factoring is quicker: try the Factoring Calculator.
• To simplify the radical you are left with at the final step, use the Square Root Calculator.
• For lines rather than parabolas, the Point-Slope Form Calculator and the Slope Calculator cover the linear equivalents.
• To tidy the fractions that appear along the way, use the Fraction Calculator.
• For general arithmetic while checking your work, the Scientific Calculator is on hand.

Sources and method

Every number on this page comes from deterministic algebra, not from external data, so no outside source is required or cited. The tool applies one exact identity, ax² + bx + c = a(x + b/2a)² + (c − b²/4a), which is proved by expanding the right-hand side: a(x + b/2a)² + c − b²/4a = ax² + bx + b²/4a + c − b²/4a = ax² + bx + c. Coefficients are stored as exact rational numbers (numerator and denominator reduced by the greatest common divisor), radicals are simplified by removing perfect-square factors, and roots follow from the square root property: if u² = v with v greater than or equal to 0 then u = ±√v, and if v is negative then u = ±i√(−v). Each result is verified by expanding the completed form back to standard form before it is displayed, so a mismatch would be reported rather than shown as correct.