Completing the Square Definitions & Examples

In geometry, completing the square is a technique used to transform certain equations into simpler forms. The general form of a quadratic equation is ax^2 + bx + c = 0. Completing the square is a way of transforming this equation into the form (x-h)^2 = k, which is known as the quadratic equation in standard form. There are a few reasons why you might want to complete the square. First, it can be used to solve equations that cannot be solved by factoring. Second, it can be used to find the center and radius of a circle when given an equation in standard form. Finally, it can be used to graph quadratic equations by graphing the corresponding circle. The process of completing the square can be broken down into four steps: 1) Identify the coefficient of x^2 and halve it. 2) Square this number and add it to both sides of the equation. 3) Factor out the leading term on each side of the equation. 4) Add or subtract the same number to both sides of the equation so that one side is equal to zero.

What is Completing the Square?

Completing the square is a mathematical technique used to solve certain quadratic equations. The general form of a quadratic equation is ax^2 + bx + c = 0. To solve a quadratic equation using the completing the square method, you need to take the following steps:

1. Rewrite the equation in standard form (ax^2 + bx = c).

2. Divide each term by a, the coefficient of x^2.

3. Add (b/2a)^2 to each side of the equation.

4. Take the square root of each side of the equation.

5. Solve for x.

Completing the Square Method

The Completing the Square Method is a mathematical technique used to solve equations. The method is used to transform an equation into what is known as quadratic form, which is easier to work with and manipulate.

To complete the square, one takes the standard form of a quadratic equation and rewrites it so that the middle term is zero. This can be done by adding to both sides of the equation or by subtracting from both sides of the equation. The process of completing the square always produces a perfect square trinomial on one side of the equation and a constant on the other side. When solving equations, this allows one to take advantage of the fact that perfect squares can be easily factored and manipulated.

The following example illustrates how the Completing the Square Method can be used to solve an equation:

Given the equation: x^2 + 6x + 9 = 0

Add to both sides: x^2 + 6x + 9 + 9 = 9

Factor the left side: (x^2 + 6x + 9) + 9 = (x^2 + 6x + 9) + (3^2)

Rewrite as: (x+3)^2 = 0+9

Solve for x: x+3=0 or -3=-3

Completing the Square Formula

The Completing the Square Formula is a mathematical formula used to solve quadratic equations. The formula is derived from the Quadratic Formula and can be used to solve any quadratic equation.

To use the formula, simply substitute the values of a, b, and c into the equation. The value of x will be equal to:

x = -b ± ?(b^2-4ac) / 2a

For example, let’s say we have the following equation:

2x^2 + 5x + 3 = 0

We can plug our values into the Completing the Square Formula and solve for x:

x = -5 ± ?(5^2-4(2)(3)) / 2(2)

x = -5 ± ?25-24 / 4

Formula for Completing the Square:

Assuming the equation is of the form:

ax^2 + bx + c = 0

The first step is to subtract c from both sides. This will put the equation in the form:

ax^2 + bx = -c

Next, divide both sides by a. This will put the equation in the form:

x^2 + (b/a)x = -(c/a)

Now, add (b/(2a))^2 to both sides. This will put the equation in the form:

(x+(b/2a))^2 = (b/(2a))^2 – (c/a)

The Steps to Completing the Square

The Steps to Completing the Square:

1. First, determine if the equation is in the form of x² + bx + c = 0. If not, rearrange it so that it is.
2. Next, take half of the coefficient of x, and square it. Add this result to both sides of the equation.
3. Now, on the left side of the equation, you will have a perfect square trinomial. On the right side, you will have a constant term.
4. To solve for x, take the square root of both sides of the equation. You should now have two equations: one with a positive value for x and one with a negative value for x. These values are your solutions!

How to Apply Completing the Square Method?

If you’re stuck on a quadratic equation with no real roots, don’t worry! You can still solve it using the completing the square method. This technique is really just a shortcut to solving equations by factoring, so if you can already do that, you can definitely complete the square.

Here’s how it works:

First, take your quadratic equation and rewrite it in standard form:

ax^2 + bx + c = 0

Next, you’ll need to find the value of “a” (the coefficient of x^2). Once you have that, divide both sides of the equation by a and simplify.

Now, take half of b (the coefficient of x) and square it. Add this new value to both sides of the equation. The left side should now look like a perfect square trinomial, and the right side will be a constant.

From here, you can solve the equation by factoring the trinomial on the left side and setting each factor equal to zero. Then just use your algebra skills to solve for x!

Example of How to Complete the Square

If you have ever completed the square on a quadratic equation, you know that it is sometimes a pain in the neck.

The process involves algebra and a little bit of geometry, but it’s not too difficult once you get the hang of it. Here is an example of how to complete the square:

Step 1: Write the equation in standard form.
This is the form of an equation where the terms are arranged so that the coefficients of the x terms are in descending order. For our equation, this would give us:

Step 2: Find one-half of the coefficient of the x term with the highest exponent. In our equation, this term is 4×2 . One-half of 4 is 2, so we’ll write 2 beside our x2 term like this:

Step 3: Square this number. Since 2 squared is 4, we’ll write 4 beside our equal sign like this:

Step 4 : Add this number to both sides of the equation. On the left side of our equation, we’ll have 4×2 +4 . On the right side, we’ll have -1+4 . This gives us:

Step 5 : Take out common factors on both sides if necessary. In our case, there is nothing to take out, so we can move on to step 6 . If there had been something to take out, we would have done that before

Why is Completing the Square Important?

The process of completing the square is important in algebra because it allows equations to be solved that cannot be solved by other means. The process of completing the square also can be used to find the center and radius of a circle when given an equation in standard form.

Conclusion

Completing the square is a mathematical process that allows you to solve certain equations by transforming them into an equivalent form that is easier to work with. This technique can be used to solve quadratic equations, factor polynomials, and find the center and radius of a circle. In this article, we’ve provided you with everything you need to know about completing the square, including step-by-step instructions and examples. We hope this article has been helpful in understanding this important mathematical concept.