### All SAT Math Resources

## Example Questions

### Example Question #1 : How To Factor A Variable

Factor the following variable

(x^{2} + 18x + 72)

**Possible Answers:**

(x + 6) (x + 12)

(x – 6) (x – 12)

(x + 18) (x + 72)

(x + 6) (x – 12)

(x – 6) (x + 12)

**Correct answer:**

(x + 6) (x + 12)

You need to find two numbers that multiply to give 72 and add up to give 18

easiest way: write the multiples of 72:

1, 72

2, 36

3, 24

4, 18

6, 12: these add up to 18

(x + 6)(x + 12)

### Example Question #2 : How To Factor A Variable

Factor 9*x*^{2} + 12*x* + 4.

**Possible Answers:**

(9*x* + 4)(9*x* – 4)

(3*x* – 2)(3*x* – 2)

(3*x* + 2)(3*x* + 2)

(9*x* + 4)(9*x* + 4)

(3*x* + 2)(3*x* – 2)

**Correct answer:**

(3*x* + 2)(3*x* + 2)

Nothing common cancels at the beginning. To factor this, we need to find two numbers that multiply to 9 * 4 = 36 and sum to 12. 6 and 6 work.

So 9*x*^{2} + 12*x* + 4 = 9*x*^{2} + 6*x* + 6*x* + 4

Let's look at the first two terms and last two terms separately to begin with. 9*x*^{2} + 6*x* can be simplified to 3*x*(3*x* + 2) and 6*x* + 4 can be simplified into 2(3*x* + 2). Putting these together gets us

9*x*^{2} + 12*x* + 4

= 9*x*^{2} + 6*x* + 6*x* + 4

= 3*x*(3*x* + 2) + 2(3*x* + 2)

= (3*x* + 2)(3*x* + 2)

This is as far as we can factor.

### Example Question #3 : How To Factor A Variable

If , and , what is the value of ?

**Possible Answers:**

–8

8

–6

0

6

**Correct answer:**

8

The numerator on the left can be factored so the expression becomes , which can be simplified to

Then you can solve for by adding 3 to both sides of the equation, so

### Example Question #4 : How To Factor A Variable

Solve for *x*:

**Possible Answers:**

**Correct answer:**

First, factor.

Set each factor equal to 0

Therefore,

### Example Question #4 : How To Factor A Variable

When is factored, it can be written in the form , where , , , , , and are all integer constants, and .

What is the value of ?

**Possible Answers:**

**Correct answer:**

Let's try to factor x^{2} – y^{2} – z^{2} + 2yz.

Notice that the last three terms are very close to y^{2} + z^{2} – 2yz, which, if we rearranged them, would become y^{2} – 2yz+ z^{2}. We could factor y^{2} – 2yz+ z^{2 } as (y – z)^{2}, using the general rule that p^{2} – 2pq + q^{2 } = (p – q)^{2} .

So we want to rearrange the last three terms. Let's group them together first.

x^{2} + (–y^{2} – z^{2} + 2yz)

If we were to factor out a –1 from the last three terms, we would have the following:

x^{2} – (y^{2} + z^{2} – 2yz)

Now we can replace y^{2} + z^{2} – 2yz with (y – z)^{2}.

x^{2} – (y – z)^{2}

This expression is actually a differences of squares. In general, we can factor p^{2} – q^{2} as (p – q)(p + q). In this case, we can substitute x for p and (y – z) for q.

x^{2} – (y – z)^{2 }= (x – (y – z))(x + (y – z))

Now, let's distribute the negative one in the trinomial x – (y – z)

(x – (y – z))(x + (y – z))

(x – y + z)(x + y – z)

The problem said that factoring x^{2} – y^{2} – z^{2} + 2yz would result in two polynomials in the form (ax + by + cz)(dx + ey + fz), where a, b, c, d, e, and f were all integers, and a > 0.

(x – y + z)(x + y – z) fits this form. This means that a = 1, b = –1, c = 1, d = 1, e = 1, and f = –1. The sum of all of these is 2.

The answer is 2.

### Example Question #4 : How To Factor A Variable

Factor and simplify:

**Possible Answers:**

**Correct answer:**

is a difference of squares.

The difference of squares formula is .

Therefore, = .

### Example Question #7 : How To Factor A Variable

Factor:

**Possible Answers:**

**Correct answer:**

We can first factor out :

This factors further because there is a difference of squares:

### Example Question #6 : How To Factor A Variable

A semi truck unloaded weighs . When the trailer compartment is loaded to half capacity with soda, the semi weighs . What will the truck weigh when the compartment is loaded to capacity with the same kind of soda?

**Possible Answers:**

**Correct answer:**

This word problem can be broken down into a basic algebraic equation:

A half-loaded semi weighs 37,000 lbs. Subtracting the weight of the truck, we can determine that a half load weighs 17,000 lbs:

From that, we can determine that a full load = 34,000 lbs:

Knowing the weight of a full load, we can calculate the weight of a 3/4 load:

Add the weight of the truck to get the total weight:

### Example Question #5 : How To Factor A Variable

A group of scientists form a global collective of temperature and climate data. U.S. temperature measurements are in Fahrenheit and must be converted to Celsius. If the average spring temperature for New York was Fahrenheit, what is the temperature value in Celsius?

The Fahrenheit to Celsius conversion equation is as follows:

**Possible Answers:**

**Correct answer:**

You can solve this problem by substituting in for and solving for :

That means that Fahrenheit is the same as Celsius.

### Example Question #6 : How To Factor A Variable

Factor to the simplest form:

**Possible Answers:**

**Correct answer:**

Group all the terms with the variable.

Pull out an term from parentheses.

There are no more common factors.

The correct answer is:

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