Completing the Square
Completing the square is a method that is used for converting a quadratic expression of the form ax^{2} + bx + c to the vertex form a(x  h)^{2} + k. The most common application of completing the square is in solving a quadratic equation. This can be done by rearranging the expression obtained after completing the square: a(x + m)^{2} + n, such that the left side is a perfect square trinomial. Completing the square method is useful in:
 Converting a quadratic expression into vertex form.
 Analyzing at which point the quadratic expression has minimum/maximum value.
 Graphing a quadratic function.
 Solving a quadratic equation.
 Deriving the quadratic formula.
Let us learn more about completing the square formula and its applications using solved examples in this page.
Completing the Square Method
The most common application of completing the square method is factorizing a quadratic equation, and henceforth in finding the roots and 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 Formula
Completing the square formula is a technique or method to convert a quadratic polynomial or equation into a perfect square with some additional constant. A quadratic expression in variable x: ax^{2} + bx + c, where a, b and c are any real numbers but a ≠ 0, can be converted into a perfect square with some additional constant by using completing the square formula or technique.
Note: Completing the square formula is used to derive the quadratic formula.
Completing the square formula is a technique or method that can also be used to find the roots of the given quadratic equations, ax^{2} + bx + c = 0, where a, b and c are any real numbers but a ≠ 0.
Formula for Completing the Square:
The formula for completing the square is: ax^{2} + bx + c ⇒ a(x + m)^{2} + n
where, m is any real number and n is a constant term.
Instead of using complex stepwise method for completing the square, we can use the following simple formula to complete the square. To complete the square in the expression ax^{2} + bx + c, first find:
m = b/2a and n = c  (b^{2}/4a)
Substitute these values in: ax^{2} + bx + c = a(x + m)^{2} + n. These formulas are derived geometrically. Let us study this in detail using illustrations in the following sections.
Completing the Square Formula Examples
Here are a few examples on the application of the completing the square formula,
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 the 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 xterm, 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 lefthand side of the equation. Simply we can replace the quadratic with the squaredbinomial form: (x  2)^{2} = 12
Now, we've completed the expression to create a perfectsquare 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.
Solving Quadratic Equations Using Completing the Square Method
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 = \(a \left(\!\!x^2+\! \frac{b}{a}x+ \!\frac{c}{a} \!\!\right) \!\!\rightarrow\!\! (1)\). 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 + (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} + bax = (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
We will observe that we will arrive at the same answer using the stepwise method also in the next section.
How to Apply Completing the Square Method?
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/2)^{2} after the 'x' term and simplify.
Trick to Learn Completing the Square Method
Here are a few tips for completing the square technique.
 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.
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Examples on Completing the Square

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 xterm, 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 = 9This process creates a quadratic expression that is a perfect square on the lefthand side of the equation. Now, we can replace the quadratic equation with the squaredbinomial form:
(x  2)^{2} = 9
Now that we have completed the expression to create a perfectsquare binomial, let us solve:
(x  2)^{2} = 9
⇒ (x  2) = ±√9
⇒ x  2 = ±3
⇒ x = 2 ± 3
⇒ x = 5, 1Answer: Using completing the square method, x = 5,  1.

Example 2: Complete the square in the quadratic expression 2x^{2} + 7x + 6.
Solution:
The given expression is 2x^{2} + 7x + 6. To complete the square, first, we will 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
Hence, the competing of square is done.
Thus,
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
We will solve by completing the square.
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 lefthand 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
FAQs on Completing the Square
What is Completing the Square Method?
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: a(x + m)^{2} + n, such that the left side is a perfect square trinomial.
What is the Easiest Way to Learn Completing the Square?
The easiest way to learn completing the square method is using the completing the square 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.
 Analyzing at which point the quadratic expression has minimum/maximum value.
 Graphing a quadratic function.
 Solving a quadratic equation.
What to Add When Completing the Square?
If we have the equation ax^{2} + bx, 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}.
How do you Complete the Square With two Variables?
To understand the completing the square method with two variables, let us see how to complete the square in the expression of 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 Completing the Square?
We use the completing 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.
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