Mục lục bài viết

“**Completing the Square**” is where we …

For those of you in a hurry, I can tell you that: and:

But if you have time, let me show you how to “**Complete the Square**” yourself.

## Completing the Square

Say we have a simple expression like x^{2} + bx. Having x twice in the same expression can make life hard. What can we do?

Well, with a little inspiration from Geometry we can convert it, like this:

As you can see x^{2} + bx can be rearranged * nearly* into a square …

… and we can **complete the square** with (b/2)^{2}

In Algebra it looks like this:

So, by adding (b/2)^{2} we can complete the square.

The result of (x+b/2)^{2} has x only **once**, which is easier to use.

## Keeping the Balance

Now … we can’t just * add* (b/2)

^{2}without also

*it too! Otherwise the whole value changes.*

**subtracting**So let’s see how to do it properly with an example:

Start with: |

Complete the Square:

(**Add and subtract** the new term)

Simplify it and we are done.

The result:

x^{2} + 6x + 7 = (x+3)^{2} − 2

And now x only appears once, and our job is done!

## A Shortcut Approach

Here is a method you may like, it is quick when you get used to it.

First think about the result we want: (x+d)^{2} + e

After expanding (x+d)^{2} we get: x^{2} + 2dx + d^{2} + e

Now see if we can turn our example into that form to discover d and e.

Example: try to fit x^{2} + 6x + 7 into x^{2} + 2dx + d^{2} + e

Now we can “force” an answer:

- We know that 6x must end up as 2dx, so
**d****must be 3** - Next we see that 7 must become d
^{2}+ e = 9 + e, so**e****must be −2**

And we get the same result (x+3)^{2} − 2 as above!

Now, let us look at a useful application: solving Quadratic Equations …

## Solving General Quadratic Equations by Completing the Square

We can complete the square to **solve** a Quadratic Equation (find where it is equal to zero).

But a general Quadratic Equation may have a coefficient of a in front of x^{2}:

ax^{2} + bx + c = 0

To deal with that we divide the whole equation by “a” first, then carry on:

x^{2} + (b/a)x + c/a = 0

## Steps

Now we can **solve** a Quadratic Equation in 5 steps:

**Step 1**Divide all terms by**a**(the coefficient of**x**).^{2}**Step 2**Move the number term (**c/a**) to the right side of the equation.**Step 3**Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p)^{2} = q, which can be solved this way:

**Step 4**Take the square root on both sides of the equation.**Step 5**Subtract the number that remains on the left side of the equation to find**x**.

## What is Completing the Square?

Completing the square is a method in algebra that is used to write a quadratic expression in a way such that it contains the perfect square. In simple words, we can say that completing the square is a process where consider a quadratic equation of the ax^{2} + bx + c = 0 and change it to write it in perfecting the square form a(x + p)^{2} + q = 0.

Completing the square method is useful in:

- Converting a quadratic expression from standard form into vertex form.
- Analyzing at which point the quadratic expression has minimum/maximum value (vertex).
- Graphing a quadratic function.
- Solving a quadratic equation.
- Deriving the quadratic formula.

Completing the square method is usually introduced in class 10. Check the following links that you may find helpful.

- Quadratic Equations Formulas Class 10
- Quadratic Equation Calculator

## Completing the Square Method

The most common application of completing the square method is factorizing a quadratic equation, and henceforth finding the roots or zeros of a quadratic polynomial or a quadratic equation. We know that a quadratic equation of the form ax^{2 }+ bx + c = 0 can be solved by the factorization method. But sometimes, factorizing the quadratic expression ax^{2 }+ bx + c is complex or NOT possible. Let us have a look at the following example to understand this case.

**For example:**

x^{2} + 2x + 3 cannot be factorized as we cannot find two numbers whose sum is 2 and whose product is 3. In such cases, we write it in the form a(x + m)^{2} + n by **completing the square**. Since we have (x + m) whole squared, we say that we have “completed the square” here. But, how do we complete the square? Let us understand the concept in detail in the following sections.

### Completing the Square Steps

To apply the method of completing the square, we will follow a certain set of steps. Given below is the process of completing the square stepwise:

- Step 1: Write the quadratic equation as x
^{2}+ bx + c. (Coefficient of x^{2}needs to be 1. If not, take it as the common factor.) - Step 2: Determine half of the coefficient of x.
- Step 3: Take the square of the number obtained in step 1.
- Step 4: Add and subtract the square obtained in step 2 to the x
^{2}term. - Step 5: Factorize the polynomial and apply the algebraic identity x
^{2 }+ 2xy + y^{2}= (x + y)^{2}(or) x^{2 –}2xy + y^{2}= (x – y)^{2}to complete the square.

## How to Apply Completing the Square Method?

Now that we have gone through the steps of completing the square in the above section, let us learn how to apply the completing the square method using an example.

**Example:** Complete the square in the expression -4x^{2} – 8x – 12.

**Solution:**

First, we should make sure that the coefficient of x^{2} is ‘1’. If the coefficient of x^{2} is NOT 1, we will place the number outside as a common factor. We will get:

-4x^{2} – 8x – 12 = -4(x^{2} + 2x + 3)

Now, the coefficient of x^{2} is 1.

- Step 1: Find half of the coefficient of x. Here, the coefficient of ‘x’ is 2. Half of 2 is 1.
- Step 2: Find the square of the above number. 1
^{2}= 1 - Step 3: Add and subtract the above number after the x term in the expression whose coefficient of x
^{2 }is 1. This means, -4(x^{2}+ 2x + 3) = -4(x^{2}+ 2x + 1 – 1 + 3) - Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the identity x
^{2 }+ 2xy + y^{2}= (x + y)^{2}. In this case, x^{2}+ 2x + 1 = (x + 1)^{2}.The above expression from Step 3 becomes: -4(x^{2}+ 2x + 1 – 1 + 3) = -4((x + 1)^{2}– 1 + 3) - Step 5: Simplify the last two numbers. Here, -1 + 3 = 2. Thus, the above expression is: -4x
^{2}– 8x – 12 = -4(x + 1)^{2}– 8. This is of the form a(x + m)^{2}+ n. Hence, we have completed the square. Thus, -4x^{2}– 8x – 12 = -4(x + 1)^{2}– 8

**Note:** To complete the square in an expression ax^{2 }+ bx + c

- Make sure the coefficient of x
^{2}is 1. - Add and subtract (b/2a)
^{2}after the ‘x’ term and simplify.

## Derivation of Completing the Square Formula

Let us complete the square in the expression ax^{2} + bx + c using the square and rectangle in Geometry. Based on the method studied earlier, the coefficient of x^{2} must be made ‘1’ by taking ‘a’ as the common factor. We get, ax^{2} + bx + c = a[x^{2} + (b/a)x + (c/a)]. Now, we will consider the first two terms, x^{2} and (b/a)x. Let us consider a square of side ‘x’ (whose area is x^{2}). Let us also consider a rectangle of length (b/a) and breadth (x) (whose area is (b/a)x).

Now, divide the rectangle into two equal parts. The length of each rectangle will be b/2a.

Attach half of this rectangle to the right side of the square and the remaining half to the bottom of the square.

To complete a geometric square, there is some shortage which is a square of side b/2a. The square of area [(b/2a)^{2}] should be added to x^{2 }+ (b/a)x to complete the square. But, we cannot just add, we need to subtract it as well to retain the expression’s value. Thus, to complete the square:

x^{2} + (b/a) x = x^{2 }+ (b/a)x + (b/2a)^{2} – (b/2a)^{2}

x^{2} + (b/a) x = x^{2} + (b/a)x + (b/2a)^{2} – b^{2}/4a^{2}

Multiplying and dividing (b/a)x with 2 gives, x^{2} + (2⋅x⋅b/2a) + (b/2a)^{2} – b^{2}/4a^{2}

By using the identity, x^{2} + 2xy + y^{2} = (x + y)^{2}

The above equation can be written as,

x^{2} + (b/a) x = (x + b/2a)^{2} – (b^{2}/4a^{2})

By substituting this in (1): ax^{2} + bx + c = a((x + b/2a)^{2} – b^{2}/4a^{2} + c/a) = a(x + b/2a)^{2} – b^{2}/4a + c = a(x + b/2a)^{2} + (c – b^{2}/4a)

This is of the form a(x + m)^{2 }+ n, where,

m = b/2a

n = c – (b^{2}/4a)

**Example:**

We will complete the square in -4x^{2 }– 8x – 12 using this formula. Comparing this with ax^{2} + bx + c, a = -4; b = -8; c = -12

Find the values of ‘m’ and ‘n’ using:

- m = b/2a = -8/2(-4) = 1
- n = c – (b
^{2}/4a) = -12 – (-8)^{2}/4(-4) = -8

Substitute these values in: ax^{2 }+ bx + c = a(x + m)^{2} + n

We get: – 4x^{2} – 8x – 12 = -4(x + 1)^{2} – 8

Note that we have already obtained the same answer by using step-wise method (not by formula) in the previous section “How to Apply Completing the Square Method?”.

**Trick to Learn Completing the Square Method**

Here are a few tips for completing the square formula.

- Step 1: Note down the form we wish to obtain after completing the square: a(x + m)
^{2}+ n - Step 2: After expanding, we get, ax
^{2}+ 2amx + am^{2}+ n - Step 3: Compare the given expression, say ax
^{2}+ bx + c and find m and n as m = b/2a and n = c – (b^{2}/4a).

**Challenging Questions:**

- Solve by completing the square: x
^{4}– 18x^{2}+ 17 = 0. Hint: Assume x^{2}= t. - Write the following equation of the form (x – h)
^{2}+ (y – k)^{2}= r^{2}by completing the square: x^{2}+ y^{2}– 4x – 6y + 8 = 0.

## Completing the Square Examples

OK, some examples will help!

Example 1: Solve x^{2} + 4x + 1 = 0

**Step 1** can be skipped in this example since the coefficient of x^{2} is 1

**Step 2** Move the number term to the right side of the equation:

**Step 3** Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.

(b/2)^{2} = (4/2)^{2} = 2^{2} = 4

**Step 4** Take the square root on both sides of the equation:

**Step 5** Subtract 2 from both sides:

And here is an interesting and useful thing.

At the end of step 3 we had the equation:

It gives us the **vertex** (turning point) of x^{2} + 4x + 1: **(-2, -3)**

Example 2: Solve 5x^{2} – 4x – 2 = 0

**Step 1** Divide all terms by 5

**Step 2** Move the number term to the right side of the equation:

**Step 3** Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation:

(b/2)^{2} = (0,8/2)^{2} = 0,4^{2} = 0,16

**Step 4** Take the square root on both sides of the equation:

**Step 5** Subtract (-0,4) from both sides (in other words, add 0,4):

**Example 1: **Using completing the square formula, find the number that should be added to x^{2} – 7x in order to make it a perfect square trinomial.

**Solution:**

The given expression is x^{2} – 7x.

**Method 1:**

Comparing the given expression with ax^{2} + bx + c, a = 1; b = -7

Using the formula, the term that should be added to make the given expression a perfect square trinomial is,

(b/2a)^{2} = (-7/2(1))^{2 }= 49/4.

Thus, from both methods, the term that should be added to make the given expression a perfect square trinomial is 49/4.

**Method 2:**

The coefficient of x is -7. Half of this number is -7/2. Finding the square,

(-7/2)^{2} = 49/4

**Example 2:** Use completing the square formula to solve: x^{2} – 4x – 8 = 0.

**Solution:**

**Method 1:**

Using formula, ax^{2} + bx + c = a(x + m)^{2} + n. Here, a = 1, b = -4, c = -8

⇒ m = b/2a = (-4)/2(1) = -2

and, n = c – (b^{2}/4a) = -8 – (-4)^{2}/4(1) = -12

⇒ x^{2} – 4x – 8 = (x – 2)^{2} – 12.

⇒ (x – 2)^{2} = 12

⇒ (x – 2) = ±√12

⇒ x – 2 = ± 2√3

⇒ x = 2 ± 2√3

**Method 2:**

Let’s transpose the constant term to the other side of the equation: x^{2} – 4x = 8. Take half of the coefficient of the x-term, which is -4, including the sign, which gives -2. Square -2 to get +4, and add this squared value to both sides of the equation:

x^{2} – 4x + 4 = 8 + 4

⇒ x^{2} – 4x + 4 = 12

This process creates a quadratic expression that is a perfect square on the left-hand side of the equation. Simply we can replace the quadratic with the squared-binomial form: (x – 2)^{2} = 12

Now, we’ve completed the expression to create a perfect-square binomial, let’s solve:

(x – 2)^{2} = 12

⇒ (x – 2) = ±√12

⇒ x – 2 = ± 2√3

⇒ x = 2 ± 2√3

**Answer:** Using completing the square method, x = 2 ± 2√3.

**Example 1:** Use completing the square method to solve: x^{2} – 4x – 5 = 0.

**Solution:**

Let us transpose the constant term to the other side of the equation:

x^{2} – 4x = 5

Now, take half of the coefficient of the x-term, which is -4, including the sign, which gives -2. Take the square of -2 to get +4, and add this squared value to both sides of the equation:

x^{2} – 4x + 4 = 5 + 4

⇒ x^{2} – 4x + 4 = 9

By using one of the algebraic identities, we can write x^{2} – 4x + 4 = (x – 2)^{2}.

(x – 2)^{2} = 9

Now that we have completed the expression to create a perfect-square binomial, let us solve:

(x – 2)^{2} = 9

⇒ x – 2 = ±3

⇒ x = 2 ± 3

⇒ x = 5, -1

**Answer:** x = 5 or -1.

**Example 2:** Complete the square in the quadratic expression 2x^{2} + 7x + 6.

**Solution:**

The given expression is 2x^{2} + 7x + 6. The first step to complete the square is to make the coefficient of x^{2} as 1. We will take the coefficient of x^{2} (which is 2) as a common factor.

2x^{2} + 7x + 6 = 2(x^{2} + (7/2)x + 3) → Equation (1)

The coefficient of x is 7/2. Half of it is 7/4. Its square is (7/4)^{2} = 49/16.

[This term can also be found using (b/2a)^{2} = [7/2(2)]^{2 }= 49/16]

Add and subtract it after the x term in Equation (1):

2x^{2} + 7x + 6 = 2(x^{2} + (7/2)x + 49/4 – 49/4 + 3)

Factorize the trinomial made by the first three terms:

2x^{2} + 7x + 6 = 2(x^{2} + (7/2)x + (49/16) – (49/16) + 3) = 2[(x + (7/4))^{2} – (49/16) + 3] = 2((x + (7/4))^{2} – (1/16)) = 2(x + (7/4))^{2} – 1/8

The final answer is of the form a(x + m)^{2} + n and hence perfecting the square has been done.

**Answer:** 2x^{2} + 7x + 6 = 2(x + (7/4))^{2} – (1/8)

**Example 3:** Solve by completing the square x^{2} – 10x + 16 = 0.

**Solution:**

The given quadratic equation is:

x^{2} – 10x + 16 = 0

Here, the coefficient of x^{2} is already 1.

The coefficient of x is (-10).

The square of half of it is (-5)^{2} = 25.

Adding and subtracting it on the left-hand side of the given equation after the ‘x’ term:

x^{2} – 10x + 25 – 25 + 16 = 0

⇒ (x – 5)^{2 }– 25 + 16 = 0 [∵ x^{2} – 10x + 25 = (x – 5)^{2}]

⇒ (x – 5)^{2} – 9 = 0

⇒ (x – 5)^{2} = 9

⇒ (x – 5) = ±√9 [Taking square root on both sides]

⇒ x – 5 = 3 OR x – 5 = -3

⇒ x = 8; x = 2

∴ x = 8, 2

**Answer:** x = 8 or 2.

## Why “Complete the Square”?

Why complete the square when we can just use the Quadratic Formula to solve a Quadratic Equation?

Well, one reason is given above, where the new form not only shows us the vertex, but makes it easier to solve.

There are also times when the form **ax ^{2} + bx + c** may be part of a

**larger**question and rearranging it as

**a(x+**makes the solution easier, because

*d*)^{2}+*e***x**only appears once.

For example “x” may itself be a function (like *cos(z)*) and rearranging it may open up a path to a better solution.

### Footnote: Values of “d” and “e”

How did I get the values of **d** and **e** from the top of the page?

Now bring everything back…

And you will notice that we have:a(x+d)^{2} + e = 0

## FAQs on Completing the Square

### What is the Method of Completing the Square?

**Completing the square** is a method in mathematics that is used for converting a quadratic expression of the form ax^{2 }+ bx + c to the vertex form a(x + m)^{2 }+ n. The most common use of this method is in solving a quadratic equation which can be done by rearranging the expression obtained after completing the square.

### What is the Easiest Way to Learn to complete the Square?

The easiest way to learn to complete the square method is using the formula, a(x + m)^{2 }+ n = a(x + m)^{2} + n. Here, m and n can be calculated with the help of the following formulas, m = b/2a and n = c – (b^{2}/4a).

### What is the Use of Completing the Square?

Completing the square formula is used for the following purposes:

- Converting a quadratic expression into vertex form.
- Computing the vertex of a quadratic function.
- Graphing a quadratic function.
- Finding the roots of a quadratic equation.

### What to Add When Completing the Square?

If we have the expression ax^{2} + bx + c, then we need to add and subtract (b/2a)^{2} which will complete the square in the expression. This will result in [x + (b/a)]^{2} – (b/2a)^{2} + c.

### How do you Complete the Square With two Variables?

Consider an expression in two variables x^{2} + y^{2} + 2x + 4y + 7. To complete the square, we take each of the coefficients of x and y, make their value half, and then square it. The coefficient of x = 2, the coefficient of y = 4. This means, (1/2 × 2)^{2} = 1 and (1/2 × 4)^{2} = 4.

Let us add and subtract this to the given equation. Then, rearrange the terms to complete the squares.

x^{2} + y^{2} + 2x + 4y + 7 + (1 – 1) + (4 – 4) = (x^{2} + 2x + 1) + (y^{2} + 4y + 4) + 7 – 1 – 4 = (x + 1)^{2} + (y + 2)^{2} + 2

### When to use Perfecting the Square?

We use the perfecting the square method when we want to convert a quadratic expression of the form ax^{2} + bx + c to the vertex form a(x – h)^{2} + k.

### What is Completing the Square Formula?

Completing the square formula is the formula required to convert a quadratic polynomial or equation into a perfect square with some additional constant. It is expressed as, ax^{2} + bx + c ⇒ a(x + m)^{2} + n, where, m and n are real numbers.

### What is the Use of Completing the Square Formula?

Completing the square formula is used when we want to represent a quadratic polynomial or equation into a perfect square with some additional constant and thus used to factorize a quadratic polynomial.

**Related Topics**

✅ Commutative Property Formula ⭐️⭐️⭐️⭐️⭐

✅ Square Root Property Formula ⭐️⭐️⭐️⭐️⭐️

✅ Root Mean Square Formula ⭐️⭐️⭐️⭐️⭐️

✅ Diagonal Of A Square Formula ⭐️⭐️⭐️⭐️⭐

✅ Perfect Square Trinomial Formula ⭐️⭐️⭐️⭐️⭐️

✅ Perimeter of a Square Formula ⭐️⭐️⭐️⭐️⭐️

✅ R Squared Formula ⭐️⭐️⭐️⭐️⭐️

✅ Regression Sum of Squares Formula ⭐️⭐️⭐️⭐️⭐️

✅ Regular Square Pyramid Formula ⭐️⭐️⭐️⭐️⭐️

✅ Secant Square x Formula ⭐️⭐️⭐️⭐️⭐️

✅ Sin squared x formula ⭐️⭐️⭐️⭐️⭐️

✅ Square Formula ⭐️⭐️⭐️⭐️⭐️

Formula for calculating the perimeter of a square