Solving by Other Methods - GED Math

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

Set each linear binomial to 0 and solve separately:

The solution set is .

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

Set each linear binomial to 0 and solve separately:

The solution set is .

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

.

Set each linear binomial to 0 and solve separately:

The solutions set is

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

.

Set each linear binomial to 0 and solve separately:

The solutions set is

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

.

Set each linear binomial to 0 and solve separately:

The solutions set is

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

.

Set each linear binomial to 0 and solve separately:

The solutions set is

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

Set each linear binomial to 0 and solve separately:

The solution set is .

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

.

Set each linear binomial to 0 and solve separately:

The solutions set is

When solving a quadratic equation, it is necessary to write it in standard form first - that is, in the form . This equation is not in this form, so we must get it in this form as follows:

We factor the quadratic expression as

so that and .

By trial and error, we find that

, so the equation becomes

Set each linear binomial to 0 and solve separately:

The solution set is .

We have our quadratic equation in the form

The quadratic formula is given as:

Using

To complete the square, we have to add a number that makes the left side of the equation a perfect square. Perfect squares have the formula .

In this case, .

Add this to both sides:

can be demonstrated to be a perfect square polynomial as follows:

It can therefore be factored using the pattern

with .

We can rewrite and solve the equation accordingly:

This is the only solution.

This quadratic equation needs to be solved by completing the square.

1. Get all of the x-terms on the left side, and the constants on the right side.

1. To put this equation into terms are more common with completing the square, we can make a coefficient of 1 in front of the term.

1. We need to make the left side of the equation into a "perfect square trinomial." To do this, we take one-half of the coefficient in front of the x, square it, and add it to both sides.

The left side is a perfect square trinomial.

1. We can represent a perfect square trinomial as a binomial squared.

1. Take the square root of both sides

1. Solve for x

Solve the equation by using the quadratic formula:

For this equation, . Plug these values into the quadratic equation and to solve for .

and

Recall the quadratic equation:

For the given equation, . Plug these into the equation and solve.

and

Solve this equation by using the quadratic equation:

For the equation ,

Plug it in to the equation to solve for .

and

The Quadratic Formula:

Plugging into the Quadratic Formula, we get

*The square root of a negative number will involve the use of complex numbers

Therefore,

To complete the square, we need to get our variable terms on one side and our constant terms on the other.

1. 

1. To make a perfect square trinomial, we need to take one-half of the x-term and square said term. Add the squared term to both sides.

1. We now have a perfect square trinomial on the left side which can be represented as a binomial squared. We should check to make sure.

* (standard form)

In our equation:

(CHECK)

1. Represent the perfect square trinomial as a binomial squared:

1. Take the square root of both sides:

1. Solve for x

or

The area of the garden is 120 square feet. The width is given by w, and the length is 2 more than 3 times the width. Going by the order of operations implied, we have length given by 3w+2.

(length) x (width) = area (for a rectangle)

In order to solve for w, we need to set the equation equal to 0.

To solve this we should use the Quadratic Formula:

(reject)

The width is 6 feet, so the length is or 20 feet.