Example Of Completing The Square

Have you ever struggled with completing the square in math class? It can be a difficult concept to grasp, but with the right approach, you'll be able to master it in no time. Check out these helpful tips and ideas on how to complete the square, using real-world examples and step-by-step instructions.

Completing the Square: Definition

What is Completing the Square?

Completing the square is a mathematical technique that is widely used in algebra. It involves the process of taking a quadratic equation and manipulating it to create a perfect square trinomial. This trinomial can then be factored easily, giving you the roots of the original equation.

How to Complete the Square

Step-by-Step Instructions

If you want to complete the square, the first step is to identify the quadratic equation that you want to work with. Once you have this equation, follow these simple steps to complete the square:

1. Move the constant term to the right-hand side of the equation
2. Divide the entire equation by the coefficient of the squared term
3. Take half of the coefficient of the x term and square it
4. Add this value to both sides of the equation
5. Factor the perfect square on the left-hand side of the equation
6. Solve for x

Let's take a look at an example of completing the square:

Example of Completing the Square

Example Problem:

Given the quadratic equation y = x² - 6x - 7, complete the square and give the solutions in exact form.

Solution:

1. Move the constant term to the right-hand side of the equation: y + 7 = x² - 6x
2. Divide the entire equation by the coefficient of the squared term: y + 7 = (x - 3)² - 9
3. Take half of the coefficient of the x term and square it: (6/2)² = 9
4. Add this value to both sides of the equation: y + 7 + 9 = (x - 3)²
5. Factor the perfect square on the left-hand side of the equation: y + 16 = (x - 3)²
6. Solve for x: x - 3 = ±√(y + 16), x = 3 + ±√(y + 16)

So the solutions in exact form are x = 3 + √(y + 16) and x = 3 - √(y + 16).

Completing the Square: Tips and Ideas

Use Completing the Square to Solve Word Problems

Completing the square can be used to solve a wide variety of problems, including word problems. Here's an example:

Suppose that you are building a rectangular fence around a garden. You have 100 feet of fencing material and you want to maximize the area of the garden. What should the dimensions of the garden be?

The first step is to write an equation for the area of the garden:

area = length x width

We know that the perimeter of the garden is 100 feet, so:

2(length + width) = 100, or length + width = 50

We can use completing the square to rewrite this equation in terms of length:

1. length + width = 50
2. length + width - 25 = 25
3. (length - 25/2)² = 625/4
4. length - 25/2 = ±√(625/4)
5. length = 25/2 + ±25/2

So the possible dimensions of the garden are length = 25 and width = 25 or length = 37.5 and width = 12.5.

Completing the Square to Solve Equations with Complex Solutions

Sometimes when you use the quadratic formula, you'll end up with a square root of a negative number. This is where completing the square can be useful. Here's an example:

Solve the equation x² + 2x + 5 = 0 for x.

The quadratic formula gives:

x = (-2 ± √(4 - 4(1)(5)))/2(1) = -1 ± i√6

This equation has complex solutions, but we can also solve it using completing the square:

1. x² + 2x + 5 = 0
2. x² + 2x = -5
3. x² + 2x + 1 = -4
4. (x + 1)² = -4
5. x + 1 = ±√(-4)

So the solutions are x = -1 + 2i and x = -1 - 2i.

Completing the square can seem intimidating at first, but with practice, you'll be able to use this technique confidently and accurately. Keep these tips in mind and don't hesitate to seek extra help if you need it!

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