What does completing the square mean?

Completing the square is a technique that rewrites a quadratic expression in the form ax squared plus bx plus c as a perfect square plus a constant: a times (x minus h) squared plus k. The resulting expression is in vertex form, which makes the vertex of the parabola at the point (h, k) immediately readable, and which can be used to solve the quadratic equation directly.

What is the formula for completing the square?

For a monic quadratic x squared plus bx plus c, add and subtract (b divided by 2) squared. This produces (x plus b over 2) squared plus c minus (b squared over 4). For a non-monic quadratic with leading coefficient a not equal to 1, first factor a out of the x squared and x terms, complete the square inside the parentheses, then redistribute. The constant adjustment is c minus b squared over 4 a.

How do you complete the square when the leading coefficient is not 1?

Factor the leading coefficient a out of the x squared and x terms first, leaving the constant outside. This produces a times (x squared plus (b over a) times x) plus c. Inside the parentheses you now have a monic expression, so apply the standard (half of the coefficient of x) squared rule. After completing the square inside, distribute the factor a back through and combine constants.

How does completing the square give the vertex form of a parabola?

Completing the square turns ax squared plus bx plus c into a times (x minus h) squared plus k, where h equals minus b over 2 a and k equals c minus b squared over 4 a. This is exactly the vertex form. The point (h, k) is the vertex of the parabola: a minimum when a is positive, a maximum when a is negative. The axis of symmetry is the vertical line x equals h.

When should I use completing the square instead of the quadratic formula?

Use completing the square when you need the vertex form for graphing, when you want to find the maximum or minimum of a quadratic, or when you are deriving the quadratic formula itself. Use the quadratic formula when you only need the roots and the coefficients are unwieldy. Both methods always work; completing the square gives more structural information, the quadratic formula is faster for just numerical roots.

Completing the Square Calculator

Key Terms

Completing the square — a technique for rewriting a quadratic expression

ax^2 + bx + c

as

a(x - h)^2 + k

, a perfect square plus a constant. Used to derive the vertex form, solve quadratics, and prove the quadratic formula.

Vertex form — the rewriting

y = a(x - h)^2 + k

of a quadratic, in which the vertex of the parabola sits at the point

(h, k)

.

Perfect square trinomial — a trinomial of the form

x^2 + 2px + p^2

that factors as

(x + p)^2

. Completing the square produces exactly this pattern inside the parentheses.

The $(b/2)^2$ rule — for the monic case

x^2 + bx + c

, the constant needed to complete the square is

(b/2)^2

. Geometrically, this is the area of the corner that turns the two half-strips into a full square.

Vertex — the turning point of a parabola:

(h, k)

in vertex form. The minimum (if

a > 0

) or maximum (if

a < 0

) point of the quadratic.

Discriminant connection — the constant adjustment after completing the square is

c - b^2/(4a)

, which leads directly to the quadratic formula and the discriminant

b^2 - 4ac

.

Getting Started

The tool opens with

x^2 + 6x + 5 = 0

loaded as a default. You see:

• A coefficient row at the top with three number inputs labeled

ax^2 + bx + c = 0

.

• A row of preset quadratics below the inputs for quick exploration.

• A geometric picture that animates a square being built piece by piece.

• A step-by-step solution panel on the right with every step rendered at once; the current step is highlighted, the others are faded.

To explore:

• Edit any of the three coefficient inputs to change the quadratic.

• Click a preset button to load a curated example.

• Press Next or the auto-play button to advance through the stages.

• Click any step on the right panel to jump the diagram back to that stage.

The final box at the bottom of the right panel always shows the resulting vertex form and the coordinates of the vertex.

Entering Coefficients

The blue input row at the top is where you set the quadratic. Three editable fields for

a

,

b

, and

c

in the standard form

ax^2 + bx + c = 0

:

• $a$ — the leading coefficient. Defaults to

1

. The tool handles any nonzero value, including negative numbers and decimals. If

a \neq 1

, an extra step appears that factors

a

out of the

x^2

and

x

terms.

• $b$ — the linear coefficient. This is the value that gets halved to find the side of the strips. The geometric diagram splits the

bx

rectangle into two equal strips of size

x \times b/2

.

• $c$ — the constant. This is the value placed into the bottom-right corner of the partial square. Whether the corner is exactly filled, partially filled (gap), or overflowed (excess) depends on how

c

compares to

(b/2)^2

.

Edits update the diagram and the step list immediately. There is no apply button.

Presets

Five preset buttons below the input row load curated quadratics chosen to show different cases of the completing-the-square procedure:

• $x^2 + 6x + 5$ — the default quadratic. Monic, positive

b

, the constant is less than

(b/2)^2 = 9

, so the diagram shows a gap.

• $2x^2 + 8x + 3$ — non-monic, with a leading coefficient of two. Triggers the extra step that factors out

2

from the

x

terms.

• $x^2 - 4x + 1$ — the negative middle term. The half-strips are oriented the same way; the sign appears in the final

(x - 2)^2

form.

• $x^2 + 5x + 2$ — the fractional half,

b/2 = 2.5

. Demonstrates that the method works for non-integer halves.

• $3x^2 + 12x + 7$ — non-monic, with a leading coefficient of three. Combines factoring and a non-trivial gap.

Click any preset to load it; the inputs, diagram, and step list all refresh.

The Default Quadratic

x^2 + 6x + 5

, frozen at its gap frame: strips of width

3

, a corner needing

9

, a constant of

5

, and a red gap of

4

.

x²3x3x5gap = 4x3x3

x² + 6x + 5, frozen at the gap

The reference preset: all integers, gap 4 a perfect square, roots −1 and −5 whole. Maximum legibility.

The default is chosen to be maximally legible: monic, everything integer, and a gap that is a perfect square, so the roots come out whole (

-1

and

-5

). It is the configuration every step section on this page freezes, which makes it the reference thread — read the step sequence with these numbers in mind and every frame belongs to one continuous story.

As a factoring cross-check:

x^2 + 6x + 5 = (x+1)(x+5)

, confirming the roots the square-completion finds. When both methods are available, they must agree — a useful habit of verification the tool makes cheap.

A Leading Coefficient of Two

2x^2 + 8x + 3

, frozen at the gap stage of its monic part

x^2 + 4x + 1.5

: strips of width

2

, a corner needing

4

, and a gap of

2.5

.

x²2x2x1.5gap = 2.5x2x2

2x² + 8x + 3, frozen at the gap

The factor of 2 ripples through: working middle term 4, half 2, and a fractional constant 1.5 in the corner.

The non-monic preset shows the division rippling through every number: the working middle term is

8/2 = 4

, the half is

2

, and the constant becomes the fraction

1.5

— integer inputs do not guarantee integer geometry once

a

is factored out.

The finish restores the factor:

2(x + 2)^2 - 5

, so the vertex is

(-2, -5)

and the gap seen in the frame gets scaled by

a

on the way back. That final multiplication is the step most often forgotten in hand calculations; the tool performs it as its own titled move — see the factoring step for the opening half of the same bracket.

The Negative Middle Term

x^2 - 4x + 1

, frozen at its gap frame: half of

-4

is

-2

, the strips carry the label

-2x

, the corner needs

4

, and the gap is

3

.

x²-2x-2x1gap = 3x-2x-2

x² − 4x + 1, frozen at the gap

Negative b lives in the labels (−2x strips), not the geometry — and resurfaces as (x − 2)² in the result.

Geometry draws lengths positive, so the strips look the same as ever — the negative sign lives in the labels and re-emerges in the algebra as

(x - 2)^2

. The picture is best read as an honest diagram of the magnitudes with the signs tracked symbolically, which is exactly how the tool renders it.

The result

(x-2)^2 - 3 = 0

gives

x = 2 \pm \sqrt{3}

— the preset where the roots go irrational, a reminder that clean inputs promise nothing about outputs. The vertex

(2, -3)

sits to the right of the axis: negative

b

pushes the parabola's turning point in the positive direction.

The Fractional Half

x^2 + 5x + 2

, frozen at its gap frame: an odd middle coefficient makes the half

2.5

, the corner

6.25

, and the gap

4.25

.

x²2.5x2.5x2gap = 4.25x2.5x2.5

x² + 5x + 2, frozen at the gap

The odd middle coefficient: strips of width 2.5, corner 6.25 — the halving trick indifferent to fractions.

The odd-

b

preset retires a common suspicion — that the halving trick only works when

b

is even. The geometry does not care: strips of width

2.5

extend the square exactly as well as integer ones, and the corner is simply

2.5^2 = 6.25

.

The completed form

(x + 2.5)^2 - 4.25

produces the roots

-2.5 \pm \sqrt{4.25}

. Fractions are precisely why completing the square is taught as a general method rather than a factoring shortcut:

x^2 + 5x + 2

has no rational factorization, yet the procedure never notices.

A Leading Coefficient of Three

3x^2 + 12x + 7

, frozen at the gap stage of its monic part

x^2 + 4x + 7/3

: strips of width

2

, a corner of

4

, and a gap of

5/3

— drawn as its decimal

1.6667

.

x²2x2x2.3333gap = 1.6667x2x2

3x² + 12x + 7, frozen at the gap

Both complications at once: factor 3 out, constant 7/3 in — leaving the thinnest red sliver of the five presets.

The last preset compounds both complications: factoring out

3

and a fractional constant

7/3

. The frame's proportions tell the story at a glance — the green block fills most of the corner, leaving a thin red sliver, the smallest gap of the five presets.

Restoring the factor gives

3(x + 2)^2 - 5

, vertex

(-2, -5)

— coincidentally the same vertex as the coefficient-two preset, reached by an entirely different quadratic. Two parabolas sharing a vertex but not a width: a closing illustration that

a

alone controls the opening, while completing the square finds the turning point.

The Geometric Diagram

The left-side Geometric Picture card animates the construction stage by stage. Each stage shows a different state of the square-in-progress:

• Start — three separate pieces appear side by side: a blue

x^2

square, an orange

bx

rectangle, and a green or red constant block for

c

.

• Factor out a (only when

a \neq 1

) — a text card noting that the rest of the work happens on the monic part; the

a

is restored at the end.

• Place the x² square — the blue square is positioned alone with both sides labeled

x

.

• Split the bx rectangle — the orange rectangle is divided into two equal strips of width

b/2

and length

x

, one placed to the right of the square and one below.

• Drop c into the corner — the green block lands in the bottom-right corner where the two strips meet. A leader line points to a label noting that the corner needs

(b/2) \times (b/2) = (b/2)^2

to be complete.

• The gap or excess — if

c < (b/2)^2

, a pulsing red block highlights the missing area equal to

(b/2)^2 - c

. If

c > (b/2)^2

, an overflow note appears instead.

• Complete the square — the full

(x + b/2) \times (x + b/2)

square is shown with dimension bars on the outside and the final vertex-form equation below, ready for the solving step.

The Starting Equation

Step one lays the raw material side by side: the blue

x^2

square, the orange

6x

rectangle, and the green constant

5

— three algebraic terms drawn as three areas, not yet arranged into anything.

x²6x5

Step 1: starting equation, frozen

Three terms as three areas: the x² square, the 6x rectangle, the constant block. Nothing arranged yet — the raw material.

The frame establishes the tool's central translation: terms are areas.

x^2

is literally a square of side

x

,

6x

a rectangle with sides

6

and

x

, and the constant a block of fixed area. Everything the procedure does from here is rearrangement — no area is ever created or destroyed, which is why the algebra stays honest.

The color code set here persists through every later frame: blue for the square, orange for anything built from the

x

-term, green for the constant, and — later — red for what is missing. Follow the colors and the algebra follows.

Factoring Out the Leading Coefficient

A step that only exists when

a \neq 1

— frozen here from the preset

2x^2 + 8x + 3

. The card announces the move to the monic part

x^2 + 4x

, with the factor of

2

set aside.

Working with the monic part:x² + 4x(factor of 2 restored at the end)

Factor-out step (2x² + 8x + 3), frozen

The non-monic interlude: work moves to the monic part x² + 4x, with the factor of 2 waiting outside until the end.

The geometric construction needs a plain

x^2

square to build on, so a non-monic quadratic is first divided down:

2x^2 + 8x

becomes

2(x^2 + 4x)

, and the picture works inside the parentheses. The factor waits outside and is restored in the final step — which is why this frame is deliberately text-only, a bookkeeping interlude between two geometric acts.

Note what changed and what did not: the working middle coefficient is now

b/a = 4

, so all later half-splitting uses

2

, not

4

— the halving happens after the factoring. Skipping this order is the classic non-monic error, and the tool structurally prevents it.

Placing the x² Square

The construction begins: a single blue square, sides labeled

x

, area

x^2

. Nothing else on the canvas.

x²xx

Step 2: the x² square, frozen

The construction site: one blue square, both sides labeled x. Everything that follows grows it symmetrically.

Starting from the square alone makes the goal visible by its absence: the finished figure will be a larger square, and everything that follows is about growing this one symmetrically. The two labeled sides matter — both must grow by the same amount for the result to stay square.

This is also where the geometric method quietly commits to

x > 0

pictures: lengths are drawn positive. The algebra it encodes has no such restriction — a point the negative-

b

preset returns to.

Splitting the bx Rectangle

The

6x

rectangle is cut into two equal strips of size

3 \times x

, one laid along the square's right edge, one along its bottom. The figure is now an L-shape — a square with two arms.

x²3x3xx3x3

Step 3: splitting 6x, frozen

Two equal 3×x strips, right and below: the halving that makes the figure a near-square instead of a rectangle.

The split in half is the step where completing the square actually happens, conceptually: only an even division lets the two strips extend the square symmetrically in both directions. Any other cut would produce a rectangle, not a near-square. Half of

6

is

3

, and that

3

is the number that will appear inside the parentheses of

(x+3)^2

.

Algebraically the frame reads

x^2 + 2 \cdot 3x

— the rewriting of

6x

as twice

3x

that seems pointless on paper and is self-evident in the picture: two identical orange strips.

The Constant in the Corner

The dashed outline marks what the L-shape is missing: a

3 \times 3

corner. The green constant

5

is dropped into it — and visibly fails to fill it, since the corner needs area

9

.

x²3x3x5corner needs3 × 3 = 9x3x3

Step 4: the constant in the corner, frozen

The dashed corner needs 3 × 3 = 9; the green 5 sits inside and visibly falls short. The mismatch is the next step’s subject.

This is the frame where the question of the whole method crystallizes: *does the constant complete the square?* The leader line does the arithmetic in place — the corner needs

(b/2)^2 = 9

— while the green block supplies only

5

. The mismatch is no failure; measuring it is precisely the next step.

The three possible answers (too little, too much, exactly right) are the three characters a quadratic can have here, and the presets cover them. For the default, the answer is "too little by 4" — see the gap step.

The Gap in the Corner

The red block makes the shortfall explicit: the corner needs

9

, the constant gives

5

, and the missing area — the gap — is

4

. The near-square is complete except for this red rectangle.

x²3x3x5gap = 4x3x3

Step 5: the gap, frozen

The red block measures the shortfall: gap = 9 − 5 = 4. The expression is exactly 4 less than a perfect square.

The gap is the number the whole procedure was built to find: the original expression equals the completed square minus the gap,

x^2 + 6x + 5 = (x+3)^2 - 4

. In the live tool this block pulses; frozen here, it simply sits where the missing area is, labeled with its value.

Its sign carries the geometry of the parabola: a positive gap means the vertex lies below the

x

-axis by exactly that amount (for monic quadratics,

k = -\text{gap}

), which is why the equation

(x+3)^2 = 4

will have two real solutions. The overflow case — constant larger than the corner — flips every one of those statements.

Reading Off the Vertex Form

The finished figure: a full square of side

x + 3

, dimension bars along both edges, the corner filled, and the equation below —

(x + 3)^2 - 4 = 0

.

x²3x3x54x +3x +3(x + 3)² − 4 = 0

Step 6: vertex form, frozen

Dimension bars read the side as x + 3, so the figure IS (x+3)² − 4 — the vertex form as a picture.

The dimension bars are the payoff of the whole construction: the big square's side is readable as

x + 3

directly from the picture, so its area is

(x+3)^2

with no expansion needed. Subtract the red gap and the vertex form is simply what the figure says.

From here the vertex coordinates fall out:

h = -3

(the sign flips because the form is

(x - h)^2

) and

k = -4

. The tool's blue vertex box activates at this step with exactly those numbers — geometry and formula agreeing in public.

Solving for x

The final step keeps the completed-square picture and turns it into roots: move the

4

across, take square roots, and

x = -3 \pm 2

— the solutions

-1

and

-5

.

x²3x3x54x +3x +3(x + 3)² − 4 = 0 → solve for x

Step 7: solving, frozen

The completed square turned into roots: (x+3)² = 4, so x = −3 ± 2 — the parabola’s symmetry made algebraic.

The solving step is deliberately anticlimactic — the hard work was geometric, and what remains is two lines of algebra on the vertex form:

(x+3)^2 = 4

, so

x + 3 = \pm 2

. The

\pm

is where the parabola's symmetry re-enters: two points at equal distance from the axis

x = -3

.

The same two lines run in general as

x = h \pm \sqrt{-k/a}

, and the sign of

-k/a

decides everything: positive gives two roots, zero one repeated root, negative a complex pair — the tool prints the

i

explicitly in that case. Completing the square doesn't just solve the equation; run on the general coefficients, it derives the quadratic formula, which is the deepest fact on this page.

The Step-by-Step Panel

The right pane lists every step of the solution at once. Each step has:

• A number in a blue circle on the left.

• A title like Split the bx rectangle or Vertex form.

• A short explanation of what is happening at this stage and why.

• A boxed math line showing the algebraic form of the expression at this stage.

The current step is fully visible with a thicker blue accent stripe; the other steps are faded and slightly blurred so the focus stays on what is happening now. The right pane auto-scrolls to keep the current step centered.

Clicking any step jumps both the diagram and the panel back to that stage. This makes it easy to scrub through the procedure: load a preset, click around the steps, study the geometric meaning of each algebraic move.

Animation Controls

Below the diagram, a strip of controls drives the animation:

• ← Back — go to the previous step. Disabled at the first step.

• Next → — advance one step. Becomes Done &check; at the last step.

• ▶ Play / ⏸ Pause — auto-advance through every step. The autoplay interval is about 2.4 seconds per step. Hitting Play after Done restarts from step 1.

• ↺ Restart — reset to step 1 without losing the current quadratic.

• Progress pips — one tick per step. The current pip is highlighted blue and stretched wider; completed pips are lighter blue. Click any pip to jump directly to that step.

The controls don't affect the coefficients — only which step is being shown. Changing the coefficients automatically resets to step 1 and stops any autoplay.

The Vertex Form Box

At the bottom of the right pane, a blue gradient box always displays the final vertex form for the current quadratic:

y = a(x - h)^2 + k

with the vertex coordinates

(h, k)

shown in yellow accent text below. This box is faded until you reach the last step, then it activates at full opacity to mark completion.

What the numbers mean for the original parabola:

• $h$ is the

x

-coordinate of the vertex: the value of

x

where the parabola turns around.

• $k$ is the

y

-coordinate of the vertex: the minimum value of the quadratic if

a > 0

, the maximum if

a < 0

.

• $a$ controls the width and direction: positive

a

opens upward, negative

a

opens downward, larger

|a|

makes a narrower parabola.

For solving the equation

ax^2 + bx + c = 0

, set the vertex form to zero and isolate

x

, which gives

x = h \pm \sqrt{-k/a}

when

-k/a \geq 0

.

What Is Completing the Square

Completing the square is a technique for rewriting a quadratic expression so it contains an explicit perfect square. The general transformation is:

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + c - \frac{b^2}{4a}

The geometric intuition the tool visualizes: think of

x^2

as a square with side

x

, and

bx

as a rectangle with sides

x

and

b

. Cut the rectangle into two equal strips of width

b/2

, place one to the right of the square and one below. The figure is almost a larger square of side

x + b/2

— it's missing one small corner of area

(b/2)^2

. Adding and subtracting

(b/2)^2

completes the picture.

Completing the square is the foundation of the quadratic formula: applying the procedure to the general

ax^2 + bx + c = 0

produces

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

in a few algebraic steps.

For deeper coverage, see the completing the square section in the quadratics theory pages.

The (b/2)^2 Step

The central trick of the method is adding

(b/2)^2

to complete the square inside the parentheses. Why this specific number?

Compare

x^2 + bx + ?

to the expansion

(x + p)^2 = x^2 + 2px + p^2

. Matching the linear terms:

2p = b

, so

p = b/2

. Matching the constant: the constant must be

p^2 = (b/2)^2

.

So

(b/2)^2

is the unique value that makes the trinomial a perfect square. If the original expression has a different constant

c

, you account for the gap or excess:

x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 + \left(c - \frac{b^2}{4}\right)

For the general

ax^2 + bx + c

, first factor

a

out of the

x^2

and

x

terms to get a monic expression inside parentheses, complete the square there, then redistribute. The constant adjustment becomes

c - b^2/(4a)

, which is exactly the

y

-coordinate of the parabola's vertex.

Related Concepts

Quadratic formula — derived directly by completing the square on the general

ax^2 + bx + c = 0

. Gives the roots

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

.

Discriminant — the expression

b^2 - 4ac

that appears under the square root in the quadratic formula. Determines whether the roots are real and distinct, repeated, or complex.

Vertex form — the rewriting

y = a(x - h)^2 + k

that completing the square produces. Makes the vertex and axis of symmetry immediately readable.

Standard form — the alternative

y = ax^2 + bx + c

. Useful for identifying intercepts and applying the quadratic formula, less convenient for graphing the vertex.

Factored form —

y = a(x - r_1)(x - r_2)

, where

r_1

and

r_2

are the roots. Useful when the quadratic factors over the rationals; not always available.

Parabola — the graph of a quadratic. The vertex form makes its shape, vertex, and orientation directly visible.

Algebra calculator — for symbolic completion of the square on more complex expressions, see the dedicated completing the square calculator in the algebra calculators section.