#### Need Help?

Get in touch with us

# Completing the Square

## Key Concepts

• Completing Square – Perfect Square Trinomial.
• Understand Vertex Form.

### Introduction

In the previous session, we have learned about solving quadratic equations using square roots and converting the equation into the form x2=d, and taking square root on both sides to find the solution.

Now we will learn about completing the square.

## Trinomial

• What is a trinomial?

A trinomial is a polynomial that consists of three terms.

Example:

1. ax2+bx+c,
1. x+y+z.
• How can you make a quadratic equation into a perfect square trinomial?

Let us see some examples.

### Perfect Square Trinomial

Example 1:

Find the value of ‘c’ that makes the expression x2+4x+c a perfect square trinomial.

Solution:

Given expression x2+4x+c

Let us use the algebraic tiles method to solve this.

The algebraic tiles for x2+4x are:

Now we re-arrange the tiles to make it a square x2+ 4x.

We need to find the value of ‘c’ such that it completes the square.

Since 4 / 2 =2 and 22 = 4

There are four tiles.

Now we convert x2+4x+4 into binomial squared (x+2)2

So, here the ‘c’ value is 4, which makes the expression a perfect square trinomial.

The process of making (b / 2)2 as ‘c’ value for the expression ax2 +bx+c is called completing square.

Example 2:

Find the value of ‘c’ that makes the expression x2+12x+c a perfect square trinomial.

Solution:

Given expression x2+12x+c

Let us use the algebraic tiles method to solve this.

The algebraic tiles for x2+12x are:

Now we re-arrange the tiles to make it a square for x2+12x.

We need to find the value of ‘c’ such that it completes the square

Since 12 / 2 =6 and 62 =36

There are four tiles.

Now we convert x2+12x+36 into binomial squared (x+2)2

So, here the ‘c’ value is 4, which makes the expression a perfect square trinomial.

Example 3:

Find the solutions for the equation x2−10x+15 = 0.

Solution:

Given equation x2−10x+15=0,

Rewrite the given equation in the form ax2+bx = d

We get,  x2−10x = −15

Add 25 on both the sides, as (10 / 2)2 = 25,  we get

10x+25 = -15+25

x²-10x+25 = 10

(x-5)² = 10

Take square root on both the sides, we get

√(x-5)² = ±√/10

(x-5) = ±√/10

Add 5 on both the sides, we get

x-5+5=5± √10

x= 5±√/10

The solutions are 5+ √10 and 5-√10

Example 4:

Find the solutions for the equation x2+8x-9 = 0

Solution:

Given equation x2+8x-9 = 0,

Rewrite the given equation in the form ax2+bx=d

We get,

x2+8x = 9

Adding 16 on both the sides, as (8 / 2)2 = 16,

we get, x²+8x + 16 = 9+16

x²+8x+16= 25

(x+4)² = = 25

Take square root on both the sides, we get

√(x+4)² = ±√/25

(x+4) = ±√/25

Subtracting 4 on both the sides, we get

x+4-4=±√25 +4

x = ±5+4=-1, 9

The solutions are -1 and 9.

### Vertex Form

An equation y = a(x−h)2−k is called the vertex form of quadratic equation, where (h, k) is the vertex.

Vertex form is used in completing square a(x−h)2

Example 1:

Find the solutions for the equation y = x2+2x−6  using vertex form.

Solution:

Given the quadratic equation, y = x2+2x−6

Now we isolate constants to the other side,

y+6 = x²+2x

Adding 1 on both sides, we get

y+6+1 = x²+2x+1

y+7= (x + 1)²

y = (x+1)² -7

The vertex form of the quadratic equation is y = (x+1)2 -7.

Example 2:

Find the solutions for the equation y = 2 +14+25 using vertex form

Solution:

Given the quadratic equation, y = x² + 14x+25

Now we isolate constants to the other side,

y-25 = x² + 14x

Adding 49 on both sides, we get

y-25+49 = x² + 14x + 49

y-25+49 (+7)² y = (x+7)2-24

The vertex form of the quadratic equation is y = (+7)² – 24.

### Vertex Form When a≠1

An equation y = a(x−h)2−k is called Vertex form of quadratic equation, where (h, k) is vertex.

Vertex form is used in completing square a(x−h)2

We have seen the cases when a = 1, now we will see the case when a≠1

Example1:

Find the solutions for the equation y = 5x2+10x+1.

Solution:

y = 5x²+10x+1

y-1 = 5x²+10x

y-1 = 5(x²+2x)

Adding (2/2)2 on both the sides,

y-1+5(1) = 5(x² + 2x+1)

y+4 = 5(x+1)²

y = 5(x+1)² – 4

### Real-Life Example

In a football match, the equation of the kick is recorded as 𝒚 = 𝟒𝒙𝟐+𝟏𝟔𝒙+𝟓. How far the ball went after the kick.

Given equation, y = 4x²+16x+5

y-5 = 4x²+16x

y-5 = 4(x² + 4x)

Adding (4/2)2 on both the sides,

y−5+4(4) = 4(x2+4x+4)

y+11 = 4(x+2)2

The height of the foot after the kick is 11 units and the football reaches the distance of 2 units from the kick point.

## Exercise

1. The coordinates of the vertex of the parabola, whose equation is y = 2x² + 4x – 5 are: a. (2,11) b. (-1,-7) c. (1,1) d. (-2,-5).
2. Find the value of ‘c’ that makes the expression x² + 8x + c a perfect square trinomial.
3. Find the solutions for the equation x² – 10x + 12 = 0.
4. Find the solutions for the equation x²+6x-9= 0
5. Find the solutions for the equation y = x²+4x-6 using vertex form.
6. Find the solutions for the equation y = x²+16x+25 using vertex form.
7. Find the solutions for the equation y = 6x2 + 24x + 2.
8. In a football match, the equation of the kick is recorded as y = 5x²+10x + 5. How far the ball went after the kick.
9. If the average of the roots is 3 and the difference is 2, find the quadratic equation.
10. The coordinates of the vertex of the parabola whose equation is y = x + 4x – 2 are:
1. (2,11)
2. (-1,-7)
3. (1,1)
4. (-2, -6)

### What have we learned

• Solving quadratic equations using completing square method.
• Perfect Square Trinomial.
• Vertex Form.

#### Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […] #### Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […] #### Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]   