# GED Math : Quadratic Equations

## Example Questions

### Example Question #51 : Quadratic Equations

Factor completely:

Explanation:

The greatest common factor of the three terms is , so factor it out, and distribute it out, as follows:

The trinomial  might be factored out as the product of two binomials , where  and  have sum 7 and product 12; by trial and error, we find that two such integers do exist, and they are 3 and 4. Therefore,

,

the correct factorization.

### Example Question #52 : Quadratic Equations

Expand the following expression:

Explanation:

Expand the following expression:

Do you recall FOIL? It is an acronym to help us remember to multiply each pair of terms when multiplying binomials.

First: Multiply the pair of first terms in this case 6x and 9x

Outer: Multiply the terms furthest to the left and right. In this case 6x and 3

Inner: Multiply the middle two terms -4 and 9x

Last: The last term from each group, in this case -4 and 3

Now, we just need to put the terms together:

And we have:

### Example Question #53 : Quadratic Equations

Multiply:

Explanation:

Take the product of the two binomials  using the FOIL method.

First:

Outer:

Inner:

Last:

Collect the like terms in the middle by subtracting coefficients:

Therefore,

Distribute the  by multiplying it by each term:

,

the correct product.

### Example Question #54 : Quadratic Equations

Expand the following expression:

Explanation:

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:

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

### Example Question #55 : Quadratic Equations

Which of the following is equivalent to ?

Explanation:

Start by FOILing.

First:

Outer:

Inner:

Last:

Combine the terms:

Finally, simplify by combining like terms.

### Example Question #56 : Quadratic Equations

Which of the following expressions is equivalent to ?

Explanation:

You must FOIL the two terms.

Now, combine like terms.

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

### Example Question #57 : Quadratic Equations

Simplify .

Explanation:

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

.

### Example Question #58 : Quadratic Equations

Foil the two equations:  and

Explanation:

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.

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.

### Example Question #59 : Quadratic Equations

Foil these two equations:  and

Explanation:

In order to foil these two equations, we're going to have to multiply them together. In order to do that, you're going to have to take each term from the first equation and separately multiply them with all the terms in the second equation.

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

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

Now we'll work with the .

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

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

We won't do this again with the second equation as that will just give us the same answers. Now take all the answers and string them together into an equation like so:

The  and  can be combined together as they share the same power, which is one.

Since there is nothing left to combine, we can leave the equation as is.

### Example Question #60 : Quadratic Equations

Foil the two equations:  and

Explanation:

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 .