Completing the Square Calculator is a free online tool that displays values for quadratic equations using the completing the square method. STUDYQUERIES’s online completing the square calculator tool makes the calculation faster, and it displays the value of the variable in a fraction of a second.

How to Use the Completing the Square Calculator?

Using the completing the square calculator is as follows:

  • Step 1: Enter the expression in the input box
  • Step 2: To get the result, click “Solve by Completing the Square”
  • Step 3: In the new window, the variable value will be displayed for the given expression

Completing The Square Calculator

What Is Completing The Square?

Completing the square is used to convert a quadratic expression of the form

$$ax^2 + bx + c$$

As a vertex

$$a(x – h)^2 + k$$

In solving a quadratic equation, completing the square is the most common application. You can do this by rearranging the expression obtained after completing the square:

$$a(x + m)^2 + n$$

Completing The Square Calculator
Completing The Square Calculator

In other words, the left side is a perfect square trinomial. Completing the square method can be useful in:

  • An expression in quadratic form can be converted into a vertex form
  • Analyzing the point at which a quadratic expression has the minimum or maximum value
  • Graphing quadratic functions
  • The solution to a quadratic equation
  • Calculating the quadratic formula

With the help of solved examples, let’s understand the completing the square formula and its applications.

Completing the Square Method

Completing the square method is often used to factor a quadratic equation, and subsequently to find its roots and zeros. In fact, quadratic equations of the form

$$ax^2 + bx + c = 0$$

The problem can be solved by factorization. But sometimes, factorizing the quadratic expression

$$ax^2 + bx + c$$

is complex or NOT possible. This can be understood by looking at the following example.

For example: $$x^2 + 2x + 3$$

As we cannot find two numbers whose sum is 2 and whose product is 3, this cannot be factored in. Instead, it is written in the following way.

$$a(x + m)^2 + n$$

by completing the square. Since we have

$$(x + m)$$

When we have the whole squared, we say that we have “completed the square.”. However, how do we complete the square? We will examine the concept in more detail below.

Completing the Square Formula

Complete the square formula is a technique or method for converting a quadratic polynomial or equation into a perfect square with some additional constant. A quadratic expression in variable x is as follows:

$$ax^2 + bx + c,\ where\ a,\ b\ and\ c\ are\ any\ real\ numbers\ but\ a \neq 0$$

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, $$ax^2 + bx + c,\ where\ a,\ b\ and\ c\ are\ any\ real\ numbers\ but\ a \neq 0$$

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.

Rather than using a complex step-by-step method to complete the square, we can use the following simple formula to complete it. To complete the square in the expression

$$ax^2 + bx + c$$

first, find:

$$m = \frac{b}{2a}\ and\ n = c – \frac{b^2}{4a}$$

Substitute these values in:

$$ax^2 + bx + c = a(x + m)^2 + n$$

The formulas are derived geometrically. Would you like to know how? This will be explained with illustrations in the following sections.

Examples To Understand Completing the Square Formula 

Here are a few examples of the application of the completing the square formula,

Example: 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,

$$m = \frac{b}{2a}^2$$

$$= \frac{-7}{2\times 1}$$

$$= 49/4$$

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,

$$\frac{-7}{2\times 1}^2 = 49/4$$

Example: Use completing the square formula to solve: $$x^2 – 4x – 8 = 0$$

Solution: Method 1:

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

$$\Rightarrow m = \frac{b}{2a} = \frac{-4}{2\times 1} = -2$$

and, $$n = c – \frac{b^2}{4a} =-8- \frac{(-4)^2}{4\times 1} = -12$$

$$\Rightarrow x^2 – 4x – 8 = (x – 2)^2 – 12$$

$$\Rightarrow (x – 2)^2 = 12$$

$$\Rightarrow (x – 2) = \pm \sqrt{12}$$

$$\Rightarrow x – 2 = \pm 2\sqrt{3}$$

$$\Rightarrow x = 2 \pm 2\sqrt{3}$$

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 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:

$$x^2 – 4x + 4 = 8 + 4$$

$$\Rightarrow x^2 – 4x + 4 = 12$$

On the left-hand side of the equation, the result is a quadratic expression that is a perfect square. The quadratic can be replaced with the squared-binomial form as follows:

$$(x – 2)^2 = 12$$

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

$$(x – 2)^2 = 12$$

$$\Rightarrow (x – 2) = \pm \sqrt{12}$$

$$\Rightarrow x – 2 = \pm 2\sqrt{3}$$

$$\Rightarrow x = 2 \pm 2\sqrt{3}$$

Answer: Using completing the square method, $$x = 2 \pm 2\sqrt{3}$$

Quick Solving Of Quadratic Equations Using Completing the Square Method

Let us complete the square in the expression

$$ax^2 + bx + c$$

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,

$$ax^2 + bx + c = a[x^2+\frac{b}{a}x+\frac{c}{a}]\longrightarrow (1)$$

Now, we will consider the first two terms,

$$x^2\ and\ \frac{b}{a}x$$

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).

Solving Quadratic Equations Using Completing the Square Method Step 1
Solving Quadratic Equations Using Completing the Square Method Step 1

Using geometry, complete the square by using a square of sides x and a rectangle of length b/a and width x.

Divide the rectangle into two equal parts now. b/2a will be the length of each rectangle.

Solving Quadratic Equations Using Completing the Square Method Step 2
Solving Quadratic Equations Using Completing the Square Method Step 2

Completing the square with geometry – The rectangle is divided into two equal parts. Each rectangle has a length of b/(2a).

Attach half of this rectangle to the right side of the square and the remaining half to the bottom.

Solving Quadratic Equations Using Completing the Square Method Step 3
Solving Quadratic Equations Using Completing the Square Method Step 3

The rectangles are rearranged so that half of the rectangles are attached to the right side of the square and the other half to the bottom of the square.

A geometric square is incomplete without a square of side b/2a. The square of area [(b/2a)²] should be added to

$$x^2 + \frac{b}{a}x$$

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:

$$x^2 + \frac{b}{a}x= x^2 + \frac{b}{a}x + \left(\frac{b}{2a} \right)^2 – \left(\frac{b}{2a} \right)^2$$

$$= x^2 + \frac{b}{a}x + \frac{b}{2a}^2 – \frac{b^2}{4a^2}$$

Multiplying and dividing $$\frac{b}{a}x$$ with 2 gives, $$x^2 + (2\times x\times \frac{b}{2a}) + \left(\frac{b}{2a} \right)^2 – \frac{b^2}{4a^2}$$

By using the identity, $$x^2 + 2xy + y^2 = (x + y)^2$$

The above equation can be written as,

$$x^2 + bax = (x + \frac{b}{2a})^2 – \frac{b^2}{4a^2}$$

By substituting this in (1): $$ax^2 + bx + c = a((x + \frac{b}{2a})^2 – \frac{b^2}{4a^2} + \frac{c}{a}$$

$$= a(x + \frac{b}{2a})^2 – \frac{b^2}{4a} + c$$

$$= a(x + \frac{b}{2a})^2 + (c – \frac{b^2}{4a})$$

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

$$m = \frac{b}{2a}$$

$$n = c – \frac{b^2}{4a}$$

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:

$$m = \frac{b}{2a} = \frac{-8}{2(-4)} = 1$$

$$n = c – \frac{b^2}{4a} = -12 – \frac{(-8)^2}{4(-4)} = -8$$

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?: Step to Step Guide

Let us learn how to apply the completing the square method using an example.

Example: Complete the square in the expression $$-4×2 – 8x – 12$$

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:

$$-4x^2 – 8x – 12 = -4(x^2 + 2x + 3)$$

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.

$$-4(x^2 + 2x + 3) = -4(x^2 + 2x + 1 – 1 + 3)$$

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, $$-4x^2 – 8x – 12 = -4(x + 1)^2 – 8$$

To complete the square in an expression $$ax^2 + bx + c$$

Make sure the coefficient of x² is 1.

Add and subtract (b/2)² after the ‘x’ term and simplify.

Short Trick to Understand Completing the Square Method

The following steps will help you learn how to apply 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 $$m = \frac{b}{2a}\ and\ n = c – \frac{b^2}{4a}$$

FAQs

How do you complete the square?

  • 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.

What is the perfect square formula?

The perfect square formula is represented in form of two terms such as $$(a + b)^2$$ The expansion of the perfect square formula is expressed as $$(a + b)^2 = a^2 + 2ab + b^2$$

Why do we complete the square?

Completing the Square is a technique that can be used to find maximum or minimum values of quadratic functions. We can also use this technique to change or simplify the form of algebraic expressions. We can use it for solving quadratic equations.

How do you complete a square with two variables?

Move all terms containing x and y to one side, and the constant term (if there is) to the other side. Divide the equation by the coefficient of x and y if it’s different from one. Complete the square in x and y. Rearrange and identify its elements.

What is another name for square root?

The term (or number) whose square root is being considered is known as the radicand.