This requires the use of the quadratic formula. Recall that:

for .

For this problem, .

So,

.

.

Therefore, the two solutions are:

Solve by factoring. We need to find two factors that multiply to eight and add to six.

One of these factors must equal zero in order for the equation to be true.

1. The first step would be to simplify, but since 2, 15, and 25 have no common factors greater than 1, simplification is impossible.

Now we factor. Multiply the first coefficient by the final term and list off the factors.

2 * 25 = 50

Factors of 50 include:

1 + 50 = 51

2 + 25 = 27

5 + 10 = 15

1. Split up the middle term to make factoring by grouping possible.

Note that the "2" and the "10," and the "5" and the "25," have to go together for factoring to come out with integers. Always make sure the groups actually have a common factor to pull.

1. Pull out the common factors from both groups, "2x" from the first and "5" from the second.

1. Factor out the "(x+5)" from both terms.

1. Set each parenthetical expression equal to zero and solve.

2x + 5 = 0, x = –5/2

x + 5 = 0, x = –5

Add 3 to both sides of the equation, to isolate the x terms:

Divide each term by 8, to isolate the x2 term:

Add the square of one half of the "b" term to each side:

Simplify:

Simplify further:

Use the following factoring rule to simplify the left side of the equation:

Take the square root of both sides of the equation:

Simplify:

Subtract one eighth from each side of the equation: or

Simplify the equations:

Solution:

The quadratic formula is as follows:

We will start by finding the values of the coefficients of the given equation:

Quadratic equations may be written in the following format:

In our case, the values of the coefficients are:

Substitute the coefficient values into the quadratic equation:

After simplifying we are left with:

leaving us with our two solutions:

and

To solve for , we can use the quadratic formula:

Write the equation in standard form by first eliminating parentheses, then moving all terms to the left of the equal sign.

First:

Inside:

Outside:

Last:

Now factor, set each binomial to zero, and solve individually. We are lookig for two numbers with sum and product ; these numbers are .

and

or

The solution set is .

To solve for x, we must first simplify the trinomial into two binomials.

To simplify the trinomial, its general form given by , we must find factors of that when added give us . For our trinomial, and the two factors that add together to get are and .

Now, using the two factors, we can rewrite as the sum of the two factors each multiplied by x:

Next, we group the first two and last two terms together and factor each of the groups:

Now, simplify further:

Finally, set each of these binomials equal to zero and solve for x:

First, we must factor out any common factors between the two terms. Both 3 and 12 share a factor of 3, so we can "take" 3 out, like this:

.

Inside the parentheses, it becomes clear that this is a difference of squares problem (a special factor), which can be solved with the equation

.

Thus, .

Now, we can set each factor to 0, and solve for :

and

.

1. After adding like terms and setting the equation equal to zero, the immediate next step in solving any quadratic is to simplify. If the coefficients of all three terms have a common factor, pull it out. So go ahead and divide both sides (and therefore ALL terms on BOTH sides) by 4.

Since zero divided by four is still zero, only the left side of the equation changes.

1. Either factor by grouping or use the square trick.

Grouping:

1 + 1 = 2

(The "1" was pulled out only to make the next factoring step clear.)

x + 1 = 0, x = –1

OR

Perfect Square:

x = –1