# Completing the Square Formula

Completing the square is converting a quadratic expression of the form ax^{2 }+ bx +c to the vertex form a(x+d)^{2} + e

Completing the square is used for:

- 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

## What is the Completing the Square Formula?

To complete the square in the expression ax^{2 }+ bx +c, first find:

d =b/2a and e = c - b^{2}/(4a)

Substitute these values in:

ax^{2 }+ bx +c = a(x+d)^{2 }+ e

## Solved Examples Using Completing the Square Formula

### Example 1:

Solve by completing the square.

x^{2}-10x+16=0

**Solution: **

The given quadratic equation is:

x^{2}-10x+16=0

We will solve this by completing the square.

Here, the coefficient of x^{2} is already 1

The coefficient of x is -10.

The square of half of it is (-5)^{2} =25

Adding and subtracting it on the left-hand side of the given equation after the x term:

x^{2 }- 10x + 25 - 25 +16=0

(x-5)^{2 }- 25 + 16 = 0

[∵ x^{2 }- 10x + 25 = (x -5)^{2} ]

(x - 5)^{2}- 9 = 0

(x - 5)^{2} = 9

(x - 5) = ±**√**9

Taking square root on both sides

x - 5 = 3; x - 5 = - 3

x = 8; x = 2

**Answer: x = 8, 2.**

### Example 2:

### What number 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:

The coefficient of x is -7

Half of this number is \(\dfrac{-7}{2}\)

Finding the square,

\[\left(\dfrac{-7}{2} \right)^2= \dfrac{49}{4}\]

Method 2:

Comparing the given expression with \(ax^2+bx+c\),

a = 1, b = -7

The term that should be added to make the given expression a perfect square trinomial is,

\[ \left( \dfrac{-b}{2a}\right)^2 = \left( \dfrac{-7}{2(1)} \right)^2= \dfrac{49}{4}\]

Thus, from both the methods, the term that should be added to make the given expression a perfect square trinomial is,

**Answer: **\(\dfrac{49}{4}\)

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