# Completing the Square — Fun with Maths

## Solve quadratic equations of the form ax² + bx + c 😎

**Completing the Square**, one of the important and easiest methods to solve a quadratic equation of the form ** ax² + bx + c. **It is an application of the famous algebraic formula

**Please check out my previous**

*(a + b)².***article**

**to understand the visual explanation for**

*(a + b)²*Why quadratic equations are that difficult to solve?🤔 Because ** x **appears twice in the equation. Is that worse? Let us consider an example equation

**and rearrange it to make**

*x2 + 2x + 1= 0*

**appear once.**

*x*How hard we try, we end up having 2 ** x**’s in our equation.

**Completing the Square **💁

Here comes our friend. Let us rearrange our equation a bit ** x² + 2x = -1** and visualize in the form of shapes

We can set aside ** -1 **(constant) and concentrate only on the LHS. The LHS consists of

1. A square with area

**(**

*x²***length =**)

*x*2. A rectangle with area

**(**

*2x***length =**

**,**

*x***breadth =**)

*2*The dashed line in the rectangle indicates that we can divide the rectangle into 2 halves. Now have

1. A Square with area ** x²** (

**length =**)

*x*2. 2 Rectangles with area

**(**

*x***length =**,

*x***breadth =**) each.

*1*Now, we rearrange our shapes to form a single square

Along with our Square (** x²**) and two Rectangles (

**), we also get an unwanted square (**

*1x + 1x = 2x***). So our equation becomes**

*1² = 1*** x2 + 2x +1 = (x + 1)² **according to

*a² + 2ab + b² = (a + b)²*To remove the unwanted square with area ** 1**, we subract the above equation with

**on either side**

*1**x² + 2x = (x + 1)² - 1*

By substitute the above value in our actual equation ** x² + 2x = -1** we get

** (x + 1)² - 1 = -1**, by sending

**to the RHS and taking square root we come up with**

*-1*** x = -1 **which solves the equation

*x² + 2x = -1*This method can be very useful in Competitive Programming. You can solve **Arranging Coins **problem in LeetCode using this method.

# Thank you 🤘

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