How to Solve Quadratic Equations by Completing the Square

Completing the Square

Completing the square is used for converting a quadratic expression of the form ax² + 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. It 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 understand the completing the square formula and its applications using solved examples in the upcoming sections.

1.	Completing the Square Method
2.	Completing the Square Formula
3.	Completing the Square Formula Examples
4.	Solving Quadratic Equations Using Completing the Square Method
5.	How to Apply Completing the Square Method?
6.	FAQs on Completing the Square

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 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 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: ax² + 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, ax² + 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: ax² + 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 ax² + bx + c, first find:

m = b/2a and n = c - (b²/4a)

Substitute these values in: ax² + bx + c = a(x + m)² + n. These formulas are derived geometrically. Are you curious to know how? We will study this geometrically in detail using illustrations in the following sections.

Completing the Square Formula Examples

Here are 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 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 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: 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.

Solving Quadratic Equations Using Completing the Square Method

Let us complete the square in the expression ax² + bx + c using 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 = \(a \left(\!\!x^2+\! \frac{b}{a}x+ \!\frac{c}{a} \!\!\right) \!\!\rightarrow\!\! (1)\). 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 + (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 + bax = (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

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 -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:

-4 x² - 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.

-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

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

• Make sure the coefficient of x² is 1.
• Add and subtract (b/2)² after the 'x' term and simplify.

Trick to Learn Completing the Square Method

Here are few steps to memorize the method of applying 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, 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.

Topics Related to Completing the Square:

• Roots Calculator
• Factorization of Quadratic Equations
• Sum and Product of Roots
• Conditions on Roots of Quadratics
• Roots of Quadratic Equation Calculator
• Nature of Roots - Examples
• Square Root

Here are some examples of completing the square.

Examples on Completing the Square

1. Example 1: Use completing the square method to solve: x² - 4x - 5 = 0.

   Solution:

   Let's transpose the constant term to the other side of the equation:

   x² - 4x = 5

   Now, 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 = 5 + 4
   ⇒ x² - 4x + 4 = 9

   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 = 9

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

   (x - 2)² = 9
   ⇒ (x - 2) = ±√9
   ⇒ x - 2 = ±3
   ⇒ x = 2 ± 3
   ⇒ x = 5, -1

   Answer: Using completing the square method, x = 5, - 1.

2. Example 2: Complete the square in the quadratic expression 2x² + 7x + 6.

   Solution:

   The given expression is 2x² + 7x + 6. To complete the square, first, we will make the coefficient of x² as 1. We will take the coefficient of x² (which is 2) as a common factor.
   2x² + 7x + 6 = 2(x² + (7/2)x + 3) → (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 (1):
   2x² + 7x + 6 = 2(x² + (7/2)x + 49/4 - 49/4 + 3)

   Factorize the trinomial made by the first three terms:

   2x² + 7x + 6 = 2(x² + (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,
   2x² + 7x + 6 = 2(x + (7/4))² - (1/8)

3. Example 3: Solve by completing the square x² - 10x + 16 = 0.

   Solution:

   The given quadratic equation is:

   x² - 10x + 16 = 0

   We will solve by completing the square.

   Here, the coefficient of x² 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:

   x² - 10x + 25 - 25 + 16 = 0
   ⇒ (x - 5)² - 25 + 16 = 0 [∵ x² - 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

4. Example 4: Which constant must be added and subtracted to solve the given quadratic equation by the method of completing the square: x² + 16x − 17 = 0?

   Solution:

   Solution by completing the square for:
   𝑥² + 16𝑥 − 17 = 0
   Keep 𝑥 terms on the left and move the constant to the right side by adding it on both sides
   𝑥² + 16𝑥 = 17
   Take half of the 𝑥 term and square it
   [16 × (1/2)]² = 64

   Answer: We need to add and subtract 64 on both sides to solve the given equation.

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Practice Questions on Completing the Square

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FAQs on Completing the Square

What is Completing the Square Method?

Completing the square is a method in mathematics used for converting a quadratic expression of the form ax² + bx + c to the vertex form a(x + m)² + n.

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 as, 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
• Analyzing at which point the quadratic expression has minimum/maximum value
• Graphing a quadratic function
• Solving a quadratic equation

What To Add While Applying Complete the Square?

Adding (b/2a)² will complete the square in the expression ax² + bx, For more information, you can go through the "Formula of Completing the Square" section of this page.

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 x² + y² + 2x + 4y + 7. To complete the square, we just make each of the coefficients of x and y half and square it first.

(2/2)² = 1

(4/2)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 Should you Apply Completing the Square?

We apply the completing 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 is the formula required to convert a quadratic polynomial or equation into a perfect square with some additional constant. It is given as,
ax² + bx + c ⇒ a(x + m)² + n, where, m and n are real numbers.

How Do You Know When to Apply Complete the Square Formula?

Completing the square formula is used to represent a quadratic polynomial or equation into a perfect square with some additional constant. It is thus used when we have to factorize any quadratic polynomial. We apply converting the square formula to convert a quadratic expression of the form ax² + bx + c to the vertex form a(x - h)² + k, we complete the square.

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.

How to Solve Quadratic Equations by Completing the Square

Source: https://www.cuemath.com/algebra/completing-the-square/

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