Completing the Square

Completing the square is a method that 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. 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.

Let us learn more about completing the square formula, its method and the process of completing the square step-wise. We will discuss its applications using solved examples for a better understanding.

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

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.

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.

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 = b/2a and
• n = c - (b²/4a)

Instead of using the complex step-wise 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 the values of m and n using the above formulas and then substitute these values in: ax² + 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 of the application of 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 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.

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.

☛ Related Articles:

• Roots Calculator
• Factorization of Quadratic Equations
• Sum and Product of Roots
• Nature of Roots - Examples

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