### All Algebra 1 Resources

## Example Questions

### Example Question #1 : How To Factor The Quadratic Equation

Factor .

**Possible Answers:**

**Correct answer:**

The expression can be factored by finding terms that multiply back to the original expression. The easiest way is to find two numbers that add to the middle term as well as multiply to the last term The numbers that satisfy both of these conditions are and , so the answer is .

### Example Question #8 : How To Factor The Quadratic Equation

Factor:

**Possible Answers:**

**Correct answer:**

Determine the signs of the binomials. Because the quadratic has a negative middle term and a positive end term, the signs will be both negative.

To factor, find the roots of that will either add or subtract to get a coefficient.

The only possibility is the set .

Substitute the roots to the factored form .

### Example Question #11 : How To Factor The Quadratic Equation

Factor this quadratic equation:

**Possible Answers:**

**Correct answer:**

Factoring is basically removing quantities from an equation. For instance has an x in each term. Therefor the fully factored form would have the most amount of x's removed from each term and would be . You cannot take out more than is present in the smallest term (smallest in the sense of what you are trying to factor out). Here we could not remove more than one x since 2x was limiting. For Quadratic equations such as we have a little easier time of factoring because we know that it will break down or factor into two binomials or two equations with two terms each. To get started recognize that since the "a" term is 1 , our first term in each binomial will just be x. Next you have to come up with two numbers that will multiply to equal the c term (the last one, 8), but add or subtract to give the middle "b" term. This comes from the fact the if you foiled your choices of your two factors the numbers must add or subtract to equal the middle term and multiply to equal the last term. Let's begin:

Start by writing two parentheses ( ) ( )

Since our "a" term is 1, the first term in each binomial will be x!

Now what numbers multiply to give the product of 8? 8 and 1. 2 and 4. Good. Which of these (if any because not every quadratic is factorable) are able to add to give the middle term of 6? Of course it is 2 and 4. So start by assuming the two factors are:

Choose an addition sign because there is no way that 2 and 4 can subtract to equal a positive 6. Now us the FOIL method to check to see if this does become when combined.

multply the first x by everything in the second parenthesis:

now mulitply 4 by every term in the second parenthesis and combine all like terms:

So are the factors of . This step by step method makes the trial and error nature of this type of factoring a little more precise. Remember, if your "a" term is one then the first term in each of your factors are x. Then just think.. "what two numbers multiply to give me the last term of my quadratic I am factoring, but also add or subtract to give the middle term.

### Example Question #51 : Quadratic Equations

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form, but this is already done.

First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The factors of are either and or and . We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, and cannot sum or difference to , but and can. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is negative in our trinomial, one and only one of our two binomials must have a negative sign. To figure out which, check the sign of in our trinomial. Since is negative, the larger of the two numbers in our binomial must be negative.

Thus, our two binomial factors are and .

### Example Question #13 : How To Factor The Quadratic Equation

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form, but this is already done.

First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The factors of are either and , and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, neither and nor and can sum to , but and can. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is positive in our trinomial, Either both signs are negative or both are positive. To figure out which, check the sign of in our trinomial. Since is negative, both signs in our binomials must be negative.

Thus, our two binomial factors are and .

### Example Question #11 : How To Factor The Quadratic Equation

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form.

--->

First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. Since the only reasonable roots of are and , we know that

Next, factor out our constant, ignoring the sign for now. The factors of are either and , and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, neither and nor and can sum to , but and can. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is positive in our trinomial, Either both signs are negative or both are positive. To figure out which, check the sign of in our trinomial. Since is negative, both signs in our binomials must be negative.

Thus, our two binomial factors are and . Note that this can also be written as , and if you graph this, you get a result identical to .

### Example Question #15 : How To Factor The Quadratic Equation

Factor the trinomial:

**Possible Answers:**

This trinomail cannot be factored because it doesn't have a middle term.

**Correct answer:**

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form, but this is already done. Note that this is a special trinomial: , and thus only and are present to be manipulated.

First, create two blank binomials.

Next, factor out our constant, ignoring the sign for now. The only factors of are either and , or and , We must select those factors which have either a difference or a sum equal to the value of in our trinomial. In this case, only and sum to , which we use since is not present.

One last step remains. We must check our signs. Since is negative in our trinomial, one sign is positive and one is negative.

Thus, our two binomial factors are and .

This is called a *difference of squares. *If you see a trinomial in the form , the roots are always and .

### Example Question #16 : How To Factor The Quadratic Equation

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form. It isn't ideal to work with , but in this equation cannot be reduced.

--->

First, create two blank binomials.

Next, factor out our constant, ignoring the sign for now. The factors of are and . Note that these do not normally sum or difference to , but the presence of as a term means one term will be doubled in value when creating . This means we either get and (no good), or and (good). Remember to add the number we want to double inside the binomial *opposite* , so it multiplies correctly. Now, we can add in our missing values:

One last step remains. We must check our signs. Since is negative in our trinomial, one sign is positive and one is negative. To figure out which, check the sign of in our trinomial. Since is positive, the bigger of our two terms **after any multipliers are completed** is the positive value. Since our terms after mutiplying are and , the must be the positive term.

Thus, our two binomial factors are and .

### Example Question #17 : How To Factor The Quadratic Equation

Factor the trinomial:

**Possible Answers:**

**Correct answer:**

To factor a trinomial without using the quadratic equation, a few basic steps can be taken. The first step is always to rearrange our trinomial into quadratic form, but this is already done. It isn't ideal to have , but we cannot reduce the constants further.

First, create two blank binomials.

Start by factoring our first term back into the first term of each biniomial. There are two valid factorings of . We could have or , and there's not really a valid system for figuring out which is correct (unless you use the quadratic formula). As a loose rule of thumb, if is near in value to , it's more likely (though by no means necessary) that the factors of are also close in value. So, let's try .

Next, factor out our constant , ignoring the sign for now. The factors of are either and , or and , but the presence of and as terms means one term will be doubled in value and the other will be tripled when creating . So, for and we either produce and (no good), or and (no good) for . If we try and with and , we get either and (no good) or and (good!) Remember to add the numbers we want to use *opposite* the term we want to combine them with , so it multiplies correctly.

Now, we can add in our missing values:

One last step remains. We must check our signs. Since is negative in our trinomial, one sign is positive and one is negative. To figure out which, check the sign of in our trinomial. Since is positive, the bigger of our two terms **after any multipliers are completed** is the positive value. Since our terms after mutiplying are and , the number that becomes must be the positive term.

Thus, our two binomial factors are and .

### Example Question #51 : Quadratic Equations

What are the roots of the following quadratic equation?

**Possible Answers:**

No solution

**Correct answer:**

Through factoring, the sum of the two roots must equal 2, and the product of the two roots must equal . , and 3 satisfy both of these criteria.

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