Completing the Square: Definition, Steps & Examples

Definitionnoun phrase

What is completing the square?

Completing the square is a method that rewrites x² + bx as a perfect square by adding (b2)²: x² + bx + (b2)² = (x + b2)². It is used to solve quadratic equations, to write a quadratic in vertex form, and to derive the quadratic formula.

In Spanish: completar el cuadrado
Also called: complete the square, completing the square method
Grade level: Algebra 1

Algebra 1Algebra 2PrecalculusCCSS HSA-REI.B.4.aCCSS HSA-SSE.B.3.bSATACTACCUPLACER

Completing the Square in a picture

x² + 6x is a square with a 3 by 3 corner missing. Adding 9 fills it, making a square of side x + 3.

Formal definition and notation

Completing the square transforms a quadratic expression ax² + bx + c into the form a(x − h)² + k, where h = −b2a and k = c − b²4a. For an equation, it produces an equivalent equation (x − p)² = q with the same solutions, which is then solved by taking square roots: x = p ± √q when q ≥ 0.

Symbol	Read it as	Meaning
(b2)²	b over 2, squared	the number that completes x² + bx to a perfect square
(x − p)² = q	x minus p, squared, equals q	the completed-square form of an equation
a(x − h)² + k	a times x minus h squared plus k	vertex form, the result for a function

Key facts

The added number is always positive, (b2)², even when b is negative: for x² − 10x, add 25.
The sign inside the binomial matches the sign of b: x² − 10x + 25 = (x − 5)².
The method works on every quadratic, including ones that do not factor.
Completing the square on ax² + bx + c = 0 in general gives the quadratic formula.
In vertex form the vertex can be read off: (x − 4)² − 5 has vertex (4, −5).

Steps to solve by completing the square

x² + bx = c ⇒ (x + b2)² = c + (b2)²

1.: If a ≠ 1, divide every term by a. Move the constant to the right side.
2.: Take half of b, square it, and add it to both sides.
3.: Write the left side as (x + b2)².
4.: Take the square root of both sides (use ±) and solve for x.

Adding the same number to both sides keeps the equation balanced. For a function, add and subtract it on the same side instead.

Completing the Square examples, step by step

Example 1: Find the missing term

Warm-up

What number completes the square for x² + 14x? Write the perfect square.

Half of 14 is 7.
Square it: 7² = 49.
x² + 14x + 49 = (x + 7)².

Answer: Add 49: x² + 14x + 49 = (x + 7)².

Example 2: Solve an equation

Core

Solve x² − 10x + 18 = 0 by completing the square.

Move the constant: x² − 10x = −18.
Half of −10 is −5; (−5)² = 25. Add 25 to both sides: x² − 10x + 25 = 7.
Write as a square: (x − 5)² = 7.
Square roots: x − 5 = ±√7, so x = 5 ± √7.

Answer: x = 5 + √7 or x = 5 − √7 (about 7.65 and 2.35).

Example 3: Vertex form with a not 1

Test-style

Write y = 2x² + 12x + 11 in vertex form and give the vertex.

Factor 2 from the x-terms: y = 2(x² + 6x) + 11.
Inside, add and subtract 9: y = 2(x² + 6x + 9 − 9) + 11.
y = 2(x + 3)² − 18 + 11.
y = 2(x + 3)² − 7.

Answer: y = 2(x + 3)² − 7, with vertex (-3, -7).

Common mistakes

✗ Mistakex² + 8x = 3. Add 16 to the left only: (x + 4)² = 3.

✓ InsteadAdd 16 to both sides: (x + 4)² = 19.

An equation stays true only if both sides change the same way.

✗ MistakeFor x² + 6x, add 6² = 36.

✓ InsteadAdd (62)² = 9: x² + 6x + 9 = (x + 3)².

Halve b before squaring. (x + 3)² has middle term 2 · 3x = 6x, which is why half of 6 is used.

✗ Mistake3x² + 12x = 9: add (122)² = 36.

✓ InsteadDivide by 3 first: x² + 4x = 3, add 4: (x + 2)² = 7.

The (b2)² rule only works when the coefficient of x² is 1.

Not to be confused with

Completing the Squarevs.Perfect square trinomial

A perfect square trinomial, such as x² + 6x + 9, is the result. Completing the square is the process of adding the 9 to make one.

Completing the Squarevs.Quadratic formula

The quadratic formula is completing the square done once in general. Completing the square works step by step on a specific equation and also gives vertex form.

Quick check: 3 questions

Question 1 of 3

Which number completes the square for x² − 18x?

Show answer and explanation

Answer: B. Half of −18 is −9, and (−9)² = 81. The added number is always positive, so −81 is wrong; 9 forgot to square; 324 squared 18 without halving.

Question 2 of 3

Completing the square turns x² + 4x − 1 = 0 into which equation?

Show answer and explanation

Answer: D. x² + 4x = 1; add 4 to both sides: (x + 2)² = 5. The first added 4 to the left only, the second used b instead of b2, and the third has the wrong sign inside.

Question 3 of 3

What is the vertex of y = x² − 6x + 2?

Show answer and explanation

Answer: A. x² − 6x + 9 − 9 + 2 = (x − 3)² − 7, so the vertex is (3, −7). (−3, −7) misread the sign, (3, 2) forgot to subtract the 9, and (6, 2) did not complete the square.

Completing the Square: frequently asked questions

Why is it called completing the square?

Because of the area picture. x² + bx can be drawn as an x by x square plus two strips of width b2, which form a square with one corner missing. The missing corner has area (b2)². Adding it literally completes the square, whose side is x + b2.

When should you complete the square instead of factoring?

Complete the square when you need vertex form, when you want the maximum or minimum of a quadratic, or when the equation does not factor and the x coefficient is even, which keeps the numbers simple. Factoring is faster when it works. The quadratic formula is often easier when a is not 1.

What if the coefficient of x squared is not 1?

Make it 1 first. For an equation, divide every term by a. For a function you want in vertex form, factor a out of the x² and x terms only, complete the square inside the parentheses, and remember that the number you subtract gets multiplied by a when it leaves the parentheses.

Practice and tools

Completing the Square CalculatorEvery step from standard form to vertex form Algebra TilesBuild the square with tiles and see the missing corner

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSA-REI.B.4.a, CCSS HSA-SSE.B.3.b

Cite this entry

Effortless Math Team. “Completing the Square.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/completing-the-square/

Flashcard

TermCompleting the Squareadd (b2)²

DefinitionAdd half of b, squared, to make a perfect square. x² + 6x + 9 = (x + 3)². Used for solving and vertex form.

Study the full deck

Browse all terms in the Math Dictionary · More algebra terms · Random term

Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Perfect SquareA perfect square is a number that equals a whole number times itself, so its square root is a whole number. For example, 49 is a perfect square because 7 × 7 = 49 and √49 = 7. The first perfect squares are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.Number System · Grade 6, Grade 7, Grade 8Vertex FormVertex form is a way to write a quadratic function as y = a(x − h)² + k, so the vertex (h, k) of its parabola can be read directly. The number a sets which way it opens and how wide it is. For example, y = 2(x − 3)² + 1 has its vertex at (3, 1).Algebra · Algebra 1, Algebra 2BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a² − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2Quadratic FormulaThe quadratic formula gives the solutions of any quadratic equation ax² + bx + c = 0 with a ≠ 0: x = −b ± √b² − 4ac2a. Substitute a, b and c, then compute once with plus and once with minus. It works even when the quadratic cannot be factored.Algebra · Algebra 1, Algebra 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s², and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2CoefficientA coefficient is the number multiplied by a variable in a term. In 4a the coefficient is 4, in −3x it is −3, and in x it is 1, because x = 1x. The sign in front belongs to the coefficient. A number standing alone, with no variable, is a constant instead.Algebra · Grade 6, Grade 7, Grade 8TrinomialA trinomial is a polynomial with exactly three terms, such as x² + 5x + 6 or 2a − 3b + 7. The terms are joined by addition or subtraction and must be unlike terms, so nothing combines. The most common kind is the quadratic trinomial ax² + bx + c, which students learn to factor into two binomial factors.Algebra · Algebra 1, Algebra 2FactoringFactoring rewrites a number or expression as a product of factors. It is the reverse of multiplying or distributing: 12 = 3 × 4, 6x + 9 = 3(2x + 3) and x² + 5x + 6 = (x + 2)(x + 3). Factoring changes the form, not the value, so multiplying the factors back gives the original.Algebra · Grade 6, Grade 7, Grade 8Quadratic EquationA quadratic equation is an equation that can be written in the form ax² + bx + c = 0, where a is not 0. Its graph is a parabola, and it may have zero, one, or two real solutions. You can solve it by factoring, by taking square roots, by completing the square, or with the quadratic formula.Algebra · Algebra 1, Algebra 2VertexA vertex is a corner point where two or more lines, rays, or sides meet. It is the point of an angle, a corner of a polygon such as a triangle, a corner of a solid, and the turning point of a parabola. The plural of vertex is vertices: a square has 4 vertices.Geometry · Grade 4, Grade 5, Grade 6Equation of a CircleThe equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r². It says that every point (x, y) on the circle is exactly r units from the center. A circle centered at the origin has the simpler equation x² + y² = r².Geometry · Geometry, Algebra 2, PrecalculusAxis of SymmetryThe axis of symmetry of a parabola is the vertical line through its vertex that splits it into two mirror-image halves. For y = ax² + bx + c, it is the line x = −b2a. Points at equal distances on either side of this line have the same y-value.Functions · Algebra 1, Algebra 2

Completing the Square

What is completing the square?

Completing the Square in a picture

Formal definition and notation

Key facts

Steps to solve by completing the square

Completing the Square examples, step by step

Example 1: Find the missing term

Example 2: Solve an equation

Example 3: Vertex form with a not 1

Common mistakes

Not to be confused with

Quick check: 3 questions

Completing the Square: frequently asked questions

Why is it called completing the square?

When should you complete the square instead of factoring?

What if the coefficient of x squared is not 1?

Related terms

Practice and tools

Sources

Flashcard

Login

Login and use all of our services.

Register

Register Fast!

Choose your next math review topic