Solve the quadratic equation by 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. 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, its method and the process of completing the square step-wise. We will discuss its applications using solved examples for a better understanding.

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 the form a(x + p)2 + q = 0. This method is generally used to find the roots of a quadratic equation. 

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 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 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:

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. 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. Let us go through them to understand the process of completing the square.

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 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 + (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

We will observe that we will arrive at the same answer using the step-wise method also in the next section.

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.)
• 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 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/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, 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

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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 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: 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 - (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 When Completing the Square?

If we have the equation ax2 + 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 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 Completing the Square?

We use 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 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.

What are Completing the Square 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.)
• 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 to complete the square.