Completing the Square Formula

“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 x² + 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 x² + 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 subtracting it too! Otherwise the whole value changes.

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:

x² + 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: x² + 2dx + d² + e

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

Example: try to fit x² + 6x + 7 into x² + 2dx + d² + 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 d² + 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 x²:

ax² + bx + c = 0

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

x² + (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 x²).
• 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 ax² + 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 ax² + bx + c = 0 can be solved by the factorization method. But sometimes, factorizing the quadratic expression ax² + bx + c is complex or NOT possible. Let us have a look at the following example to understand this case.

For example:

x² + 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 x² + bx + c. (Coefficient of x² 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 x² term.
• Step 5: Factorize the polynomial and apply the algebraic identity x² + 2xy + y² = (x + y)² (or) x^{2 –} 2xy + y² = (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 -4x² – 8x – 12.

Solution:

First, we should make sure that the coefficient of x² is ‘1’. If the coefficient of x² is NOT 1, we will place the number outside as a common factor. We will get:

-4x² – 8x – 12 = -4(x² + 2x + 3)
Now, the coefficient of x² 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. 1² = 1
• Step 3: Add and subtract the above number after the x term in the expression whose coefficient of x² is 1. This means, -4(x² + 2x + 3) = -4(x² + 2x + 1 – 1 + 3)
• Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the identity x² + 2xy + y² = (x + y)². In this case, x² + 2x + 1 = (x + 1)².The above expression from Step 3 becomes: -4(x² + 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: -4x² – 8x – 12 = -4(x + 1)² – 8. This is of the form a(x + m)² + n. Hence, we have completed the square. Thus, -4x² – 8x – 12 = -4(x + 1)² – 8

Note: To complete the square in an expression ax² + bx + c

• Make sure the coefficient of x² 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 ax² + bx + c using the square and rectangle in Geometry. Based on the method studied earlier, the coefficient of x² must be made ‘1’ by taking ‘a’ as the common factor. We get, ax² + bx + c = a[x² + (b/a)x + (c/a)]. Now, we will consider the first two terms, x² and (b/a)x. Let us consider a square of side ‘x’ (whose area is x²). 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 x² + (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:

x² + (b/a) x = x² + (b/a)x + (b/2a)² – (b/2a)²

x² + (b/a) x = x² + (b/a)x + (b/2a)² – b²/4a²

Multiplying and dividing (b/a)x with 2 gives, x² + (2⋅x⋅b/2a) + (b/2a)² – b²/4a²

By using the identity, x² + 2xy + y² = (x + y)²

The above equation can be written as,

x² + (b/a) x = (x + b/2a)² – (b²/4a²)

By substituting this in (1): ax² + bx + c = a((x + b/2a)² – b²/4a² + c/a) = a(x + b/2a)² – b²/4a + c = a(x + b/2a)² + (c – b²/4a)

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

m = b/2a

n = c – (b²/4a)

Example:

We will complete the square in -4x² – 8x – 12 using this formula. Comparing this with ax² + 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 – (b²/4a) = -12 – (-8)²/4(-4) = -8

Substitute these values in: ax² + bx + c = a(x + m)² + n

We get: – 4x² – 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, ax² + 2amx + am² + n
• Step 3: Compare the given expression, say ax² + bx + c and find m and n as m = b/2a and n = c – (b²/4a).

Challenging Questions:

• Solve by completing the square: x⁴ – 18x² + 17 = 0. Hint: Assume x² = t.
• Write the following equation of the form (x – h)² + (y – k)² = r² by completing the square: x² + y² – 4x – 6y + 8 = 0.

Completing the Square Examples

OK, some examples will help!

Example 1: Solve x² + 4x + 1 = 0

Step 1 can be skipped in this example since the coefficient of x² 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 = 2² = 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 x² + 4x + 1: (-2, -3)

Example 2: Solve 5x² – 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 = 0,4² = 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 x² – 7x in order to make it a perfect square trinomial.

Solution:

The given expression is x² – 7x.

Method 1:

Comparing the given expression with ax² + 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: x² – 4x – 8 = 0.

Solution:

Method 1:

Using formula, ax² + bx + c = a(x + m)² + n. Here, a = 1, b = -4, c = -8

⇒ m = b/2a = (-4)/2(1) = -2

and, n = c – (b²/4a) = -8 – (-4)²/4(1) = -12

⇒ x² – 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: x² – 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:

x² – 4x + 4 = 8 + 4
⇒ x² – 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.

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 ax² + bx + c may be part of a larger question and rearranging it as a(x+d)² + e makes the solution easier, because 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 ax² + 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 – (b²/4a).

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 ax² + 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 x² + y² + 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.

x² + y² + 2x + 4y + 7 + (1 – 1) + (4 – 4) = (x² + 2x + 1) + (y² + 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 ax² + 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, ax² + 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.

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