Completing the Square Calculator with steps | Complete the Square

Completing the Square Calculator

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In algebra there are several techniques to solve quadratic equations, one of the most used is the technique of completing the square. With the Completing the Square Calculator you will be able to solve all kinds of quadratic equations (solving quadratic equations by completing the square has never been so easy).

This calculator is completely free, the only thing we ask you in return is that you share it with anyone you think might be useful.

Instructions for use

To use the complete the square calculator follow these simple steps:

  1. Enter the quadratic equation you want to solve by completing the square. The equation you enter must contain only one variable.
  2. Press the blue “Calculate” button, doing so will display the detailed step-by-step solution.

What is completing the square?

Completing the square is an algebraic method with which we can convert a quadratic expression from the form ax2+bx+c to the form a(x–h)2+k, through basic arithmetic operations. In other words, Completing the Square is the process of manipulating the form of the equation so that the left side of the equation becomes a perfect square trinomial.

Completing the square is a useful technique for solving quadratic equations because it allows you to find the solutions in a simple and straightforward manner. It is also a useful tool for graphing quadratic equations because it allows you to easily identify the vertex of the graph.

How to complete the square | How to solve quadratic equations by completing the square

To solve quadratic equations by completing the square, the following procedure must be performed:

ax2+bx+c=0

  1. Manipulate the equation so that the coefficient with no variable, c, is found only on the right side of the equation. To achieve this you must add or subtract the value of c on both sides. If c has a positive sign, you must subtract c from both sides, but if its sign is negative, you must add its value to both sides.
  2. If the value of the coefficient a is different from 1, you must divide each of the terms in the equation by the value of a.
  3. Now you must add the term (b/2a)2 to both sides of the equation, with this we complete the perfect square trinomial on the left side of the equation.
  4. Factor the left side as the square of a binomial.
  5. Then find the square root of both sides and solve for x.

Completing the Square Formula

The steps explained above are summarized in the following formula:

ax2 + bx + c ⇒ a(x + m)2 + n

where,

m = b/2a and n = c – (b2/4a)

Here is an example using the completing the square calculator:

Completing Square Examples

Here are some examples generated with the help of the completing the square calculator.

Example01

Solve by completing the square: 5x2 – 13x = 5

5x2−13x=5
Step 1: Divide both sides by 5.
‌
5x2−13x
5
‌=‌
5
5
‌
x2+‌
−13
5
‌x=1
Step 2: To complete the square add 169/100 to both sides.
x2+‌
−13
5
‌x+‌
169
100
‌=1+‌
169
100
‌
x2+‌
−13
5
‌x+‌
169
100
‌=‌
269
100
‌
Step 3: Factor left side.
(x‌
−13
10
‌)2=‌
269
100
‌
Step 4: Take square root.
x‌
−13
10
‌=±√‌
269
100
‌
Step 5: Add 13/10 to both sides.
x‌
−13
10
‌+‌
13
10
‌=‌
13
10
‌±√‌
269
100
‌
x=‌
13
10
‌±√‌
269
100
‌
x=‌
13
10
‌+‌
1
10
‌√269‌‌ or ‌‌x=‌
13
10
‌+‌
−1
10
‌√269
Solution:
x=‌
13
10
‌+‌
1
10
‌√269‌‌ or ‌‌x=‌
13
10
‌+‌
−1
10
‌√269

Example02

Solve by completing the square: x2 + 6x + 13 = 0

x2+6x+13=0
Step 1: Subtract 13 from both sides.
x2+6x+13−13=0−13
x2+6x=−13
Step 2: To complete the square add 9 to both sides.
x2+6x+9=−13+9
x2+6x+9=−4
Step 3: Factor left side.
(x+3)2=−4
Step 4: Take square root.
x+3=±√−4
Step 5: Add -3 to both sides.
x+3−3=−3±√−4
x=−3±√−4

Completing the square worksheet

So that you can practice the technique of completing the square, here you can download a worksheet with exercises:


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