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Completing The Square Calculator

Complete the square of a quadratic equation and find its roots by just adding a few inputs into this online calculator.

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Use completing the square calculator to solve any given quadratic equation of the form ax² + bx + c = 0 in seconds. This handy tool uses completing the square method to solve quadratic equations and provides precise results.

What Is Meant By Completing The Square?

This is a method that is used to solve quadratic equations. It means to change an equation's left side in a way that it becomes the square of a binomial. It is a technique for converting the form of a quadratic polynomial (ax² + bx + c = 0 ) to the (x−h)^2 = k form. Completing the square method is very useful when it becomes hard to solve the equation by using quadratic formula. An equation should be in the form of an ax² + bx + c = 0 for the implementation of completing the square. 

How To Complete The Square?

Go through the following steps to complete the square of a quadratic equation:
  • The first thing that you have to do is to arrange the equation in the ax² + bx + c = 0 form.
  • Now check if “a” is not equal to 1, then divide both sides by “a”.
  • Take the term “c” to the right side of the equation by adding or subtracting on both sides.
  • Divide the term “b” by 2 and square it.
  • Now add the result on both sides
  • Write the perfect square on the left sides as (x + y)^2
  • Apply square root on both sides

Special Cases:

What If a ≠ 1?

When you see that the “a” is not equal to one, it is greater or lower than one, then use “a” to divide both sides of the equation.  Let's complete the square of an equation in which “a” is not equal to one: \[2x^2 - 6x - 5 = 0\] Now a = 2, so divide all terms by 2: \frac{2}{2}x^2 - \frac{6x}{2} - \frac{5}{2} = \frac{0}{2} \[x^2 - 3x - \frac{5}{2} = 0\] \[x^2 - 3x -  \frac{5}{2} = 0\] If it seems difficult, then see the following example that we have solved, but if you still did not get it, then solve by completing the square calculator.

What If b = 0?

If b is equal to zero, then it means you don't have the “x” term. It becomes easy to solve the equation because you have to solve for the x-squared term as we have done below: \[x^2 - 0x - 6 = 0\] \[x^2  - 6 = 0\] \[x^2 = 6\] \[x^2 = \sqrt{6}\] X = +2.4495 X = - 2.4495

Practical Example: 

Let's suppose there is a quadratic equation whose coefficients are a = 2, b = -6, and c = -10. Now how to solve by completing the square?

Solution:

Given that: \[2x^2 - 6x - 10 = 0\] Firstly, divide the entire equation by the coefficient of \(x^2\), which is 2: \[x^2 - 3x - 5 = 0\] Now add and subtract \(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\) to the left side of the equation: \[x^2 - 3x + \frac{9}{4} - \frac{9}{4} - 5 = 0\] Rearrange the equation: \[x^2 - 3x + \frac{9}{4} = \frac{9}{4} + 5\] Now, simplify the right side: \[x^2 - 3x + \frac{9}{4} = \frac{9}{4} + \frac{20}{4}\] \[x^2 - 3x + \frac{9}{4} = \frac{29}{4}\] \[(x - \frac{3}{2})^2 = \frac{29}{4}\] Take the square root: \[x - \frac{3}{2} = \pm \frac{\sqrt{29}}{2}\] Now, solve by completing the square: \[x - \frac{3}{2} = \pm \frac{\sqrt{29}}{2}\] Now, isolate \(x\): \[x = \frac{3}{2} \pm \frac{\sqrt{29}}{2}\] So, the solutions for the quadratic equation \(2x^2 - 6x - 10 = 0\) using the completing the square method are: \[x = \frac{3}{2} + \frac{\sqrt{29}}{2}\] = 4.1926 and \[x = \frac{3}{2} - \frac{\sqrt{29}}{2}\] = -1.1926 That's how you can manually solve the equation by completing the square. But if you don't have the time, then simply utilize a completing the square formula calculator. It will let you perform the calculation instantly without much manual intervention.

How Does Our Completing The Square Calculator Work?

This free complete the square calculator shows the variable value for the quadratic equation in no time by processing the entered input values. Let's see how it works!

What Do You Need To Enter?

  • Add the coefficients of your quadratic equation in the specified fields that are represented by a,b, and c.

This Is What You Will Get!

  • It provides a solution for x in the form of one or two values. These values are the roots of the quadratic equation. 
  • A step-by-step calculation is provided so that you can easily understand each step.
This complete square calculator is solely designed to make the process of solving quadratic equations by completing the square easy and quick. 

Benefits of Using a Complete The Square Formula Calculator:

Let's take a look at the following benefits of using an online complete the square calculator:
  • High-speed calculation
  • No risk of human errors
  • Free of cost
  • No registration or signup is required
  • Direct access from the browser

FAQ’s:

Why Is Completing Squares Important?

This is a technique that is used to get the maximum and minimum values of a quadratic equation. It can be used to simplify the algebraic equations. Completing the squares process solves the quadratic equations swiftly which can not be solved by the quadratic formula.  

What Are 3 Common Techniques for Solving A Quadratic Equation?

There are three methods to solve a quadratic equation and that are:
  • Factoring method
  • Completing square method
  • Quadratic formula method
The most convenient way to solve the quadratic equation is by using the completing the square calculator with steps.

References:

From the source of socratic.org: How do you solve an equation by completing the square? From the source of wikipedia.org: Completing the square. 
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