Quadratic Equations - GED Math

Expand the following expression:

This is what is known as a difference of squares. We can use FOIL to find our answer.

Remember FOIL? First Outer Inner Last

This means we need to multiply each pair of terms in parentheses to get the correct answer:

First:

Outer:

Inner:

Last:

Put it all together to get:

Making our answer

Do you see why it's called a difference of squares?

Foiling means to take two equations and merge them into one. It's also the same as saying you want to multiply one equation with another, which is what we'll be doing.

Our two equations are and , which is the same as . In order to multiply these two equations together, you must first multiply the first unit in your equation with everything in the second equation, then the second unit with everything in your second equation.

Multiply the in the first equation with the from the second equation:

Multiply the in the first equation with the from the second equation:

Now multiply the with the from the second equation:

Multiply the from the first equation with the from the second equation:

We won't multiply the second equation with the first one like we did above, as that would give us the same answers. Take all the answers you got from above and now string them together like so:

We're almost done, but we seem to be have more than one ; and . These two terms can be combined like so:

Since nothing else seems to have more than one of itself, we can now put the equation together. Make sure to go in order of highest power of to the lowest power of .

Your answer is

To foil these two equations, we'll need to multiply them together. To multiply them together, you'll have to take each term from the first equation and multiply them individually with each term in the second equation.

We'll start with from the first equation.

Multiply from the first equation with from the second equation.

Multiply from the first equation with from the second equation.

Now we'll use the from the first equation.

Multiply the from the first equation with the from the second equation.

Multiply the from the first equation with from the second equation.

We won't do this method with the second equation as that will only give us the same answer. Now take all of your answers and string them together, like so:

We can combine our and because they are under the same power of ; which is one.

Since we cannot combine anymore like terms, we can take what we have left and put it as our final equation.

Your answer is

Start by FOILing.

First:

Outer:

Inner:

Last:

Combine the terms:

Finally, simplify by combining like terms.

You must FOIL the two terms.

Now, combine like terms.

Thus, the expanded version of the given terms is the following:

This is a classic FOIL problem. FOIL stands for first, outer, inner, and last. It describes a process of multiplying together polynomials. Essentially, you are multiplying every combination of terms from the first set of parentheses and the second set of parentheses. You start with the first two terms, then the outer two terms, then the inner two terms, and finally the last two terms.

For , your first two terms are and , and . Your outer two terms are and , and . Your inner two terms are and , and . Your last two terms are and , and .

Ultimately, once you combine and add everything together, you get

.

You finish by combining like terms. The two like terms here are and and .

Therefore, your final answer is

.

Start by factoring the numerator.

To factor the numerator, you will need to find numbers that add up to and multiply to .

Next, factor the denominator.

To factor the denominator, you will need to find two numbers that add up to and multiply to .

Rewrite the fraction in its factored form.

Since is found in both numerator and denominator, they will cancel out.

Using the FOIL distribution method:

First:

Outer:

Inner:

Last:

Resulting in:

Combining like terms, the 's combine for a final answer of:

This is a FOIL-ing problem. First, set up the numbers in a form we can use to create the function.

Take the opposite sign of each of the numbers and place them in this format.

Multiply the in the first parentheses by the and 8 in the second parentheses respectively to get

Multiply the in the first parentheses by the and 8 in the second parentheses as well to give us .

Then add them together to get

Combine like terms to find the answer which is .

Simplify using FOIL method.

Remember that multiplying variables means adding their exponents.

F:

O:

I:

L:

Combine the terms. Note that we cannot simplify further, as the exponents do not match and cannot be combined.

The FOIL method yields the products below.

First:

Outside:

Inside:

Last:

Add these four terms, and combine like terms, to obtain the product of the binomials.

First, factor out an , since it is present in all terms.

We need two factors that multiply to and add to .

and

Our factors are and .

We can check our answer using FOIL to get back to the original expression.

First:

Outside:

Inside:

Last:

Add together and combine like terms.

Distribute the that was factored out first.

Using the FOIL method is simple. FOIL stands for First, Outside, Inside, Last. This is to help us make sure we multiply every term correctly looking at the terms inside of each parentheses. We follow FOIL to find the multiplied terms, then combine and simplify.

First, stands for multiply each first term of the seperate polynomials. In this case, .

Inner means we multiply the two inner terms of the expression. Here it's .

Outer means multiplying the two outer terms of the expression. For this expression we have .

Last stands for multiplying the last terms of the polynomials. So here it's .

Finally we combine the like terms together to get

.

This problem involves multiplying two binomials. To solve, we will need to use the FOIL method.

Comparing this with our original equation, , , , and .

Using these values, we can substitute for the FOIL equation.

Notice that the two center terms use the same variables; this allows us to combine like terms.

To solve, it may be easier to convert the radicals to exponents.

Remember, the method used in multiplying two binomials is given by the equation:

Comparing this with our expression, we can identify the following variables:

We can substitute these values into the FOIL expression. Multiply to simplify.

Simplify by combining like terms. The center terms are equal and opposite, allowing them to cancel to zero.

A term to a given power can be combined with another term with the same base using the identity . This allows us to adjust our final answer.

Using the FOIL distribution method:

First:

Outer:

Inner:

Last:

Resulting in:

Combining like terms, the 's cancel for a final answer of:

First:

Outside:

Inside:

Last:

Add together:

FOIL:

First:

Outer:

Inner:

Last:

Add these together and combine like terms:

is a special type of factorization.

When simplified, the "middle terms" cancel out, because they are the same value with opposite signs:

Expressions in the form always simplify to

At this point, we know that the only possible answers are q2 and -81.

However, now we have to check the terms of the second expression to see if we find any similarities.

Here we notice that rather than cancelling out, the middle terms combine instead of cancel. Also, our final term is the product of two negative numbers, and so is positive. Comparing the two simpified expressions, we find that only is shared between them.