What is the area, in square inches, of a circle with a diameter equal to 12 inches?

Solution
A circle with diameter 12 inches has a radius of 6 inches. The area of a circle is A = πr^{2}, where r is the radius. Thus, the area of this circle is A = π(6^{2})= 36π square inches.
If, a, b, and c are positive integers such that ab = m and c^{2}b = n, then mn = ?

Solution
If a^{b} = m and c^{2b} = n, then mn = (a^{b})(c^{2b}). Because a and c are not necessarily the same number, you cannot simply add the exponents. The only legitimate answer choice that further manipulates the expression (a^{b})(c^{2b}) factors out the b exponent and places it outside a set of parentheses. Thus(a^{b})(c^{2b}) = (a^{b}c^{2b}) = (ac^{2})^{b}.
In Sulema’s geography class, all tests count equally. So far, Sulema has taken 2 of the 3 tests in geography this marking period and earned scores of 88% and 79%, respectively. What is the minimum score Sulema needs on the third test to have a test average of 87%?

Solution
A good way to go about solving this problem is to think of the tests as each having a value of 100 points, making the total of all 3 tests worth 300 points. In order to average 87% overall, Sulema would then need to score 87% of the 300 points, or 300(0.87), or 261 points. By scoring 88% and 79% on her previous tests, she has already obtained 88 + 79, or 167 points. Thus Sulema must score 261 − 167, or 94 points on the third test, which is equivalent to 94%.
What are all the solutions for x if 3x^{2} − 2x − 21 = 0?

Solution
For the equation 3x^{2} − 2x − 21 = 0, all solutions of x can be found by either factoring or by using the quadratic formula \(\left [ x=\frac{(b\pm \sqrt{b^{2}− 4ac})}{2a}for quadratic equations ax^{2} + bx + c = 0 \right ]\). In this case, since the final term is negative, you know that when this equation is factored, one sign will be positive and the other will be negative. Knowing that the factors of 21 are 1 and 21 and 3 and 7, you can guess and check your answers with relative ease. In this case, the correct answer is (3x +7)(x −3) = 0. Solve for x: x = −^{7}⁄_{3} or x = 3.
The length of a rectangle is 5 inches longer than its width. If the perimeter of the rectangle is 38 inches, what is the width, in inches?

Solution
In a rectangle with length l which is 5 inches longer than the width w, the relation between the sides is l = w + 5. Since perimeter, which is the distance around, can be written as twice the length plus twice the width, 38 = 2l+2w. In order to solve for width w, use the equation l = w + 5 to substitute into the perimeter equation:
38 = 2(w + 5) + 2w
38 = 2w + 10 + 2w
38 − 10 = 4w + 10 − 10
\(\frac{28}{4}=\frac{4w}{4}\)
w = 7
While doing research on the climates of South American countries, Andrea notices that all of the temperatures are given in degrees Celsius. Because she is not as familiar with the Celsius temperature scale, it is difficult for her to know whether a location with an average temperature of 25°C has a warm climate. Fahrenheit, F, and Celsius, C, are related by the formula F =(^{9}⁄_{5})C+32. What is the temperature in degrees Fahrenheit of the location with an average temperature of 25°C?

Solution
To convert 25°C to degrees Fahrenheit using the formula F=(^{9}⁄_{5})C+32,
simply substitute 25 for C in the formula:
(^{9}⁄_{5})25 + 32 = 77°F
In the equation r = \(\frac{4}{(2 + k)}\), k represents a positive integer. As k gets larger without bound, the value of r:

Solution
As k gets larger and larger without bound, the expression \(\frac{4}{(2 + k)}\) becomes 4 divided by an increasingly large number. For example, think about the trend between the following fractions:\(\frac{4}{100},\frac{4}{10,000},\frac{4}{1,000,000}\),... Looking at it this way, you can see that the expression for r gets closer and closer to zero.
Which of the following is a polynomial factor of x^{2} − 2x − 24?

Solution
To solve this problem it is necessary to factor the polynomial x^{2} − 2x − 24. In this case, since the final term is negative, you know that when this equation is factored, one sign will be positive and the other will be negative. Further, since the term is −24, the numbers in the problem will be factors of 24. The factors of 24 are 1 and 24, 2 and 12, 3 and 8, or 4 and 6. It is logical to look for the factors that have a difference of 2 because of the −2x term in the original polynomial. In this case, 4 and 6 are the only factors of 24 that have a difference of 2. Because the term −2x is negative, the logical expression is (x + 4)(x − 6). Therefore, x + 4 is the correct answer choice.
The length of a side of a square is represented as (3x − 2) inches. Which of the following general expressions represents the area of the square, in square inches?

Solution
A square with sides (3x − 2) inches would have an area of (3x − 2)(3x − 2) square inches. Use the “FOIL” method to multiply all terms in the equation:
First: (3x)(3x) = 9x^{2}
Outside: (3x)(−2) = −6x
Inside: (−2)(3x) = −6x
Last: (−2)(−2) = 4
Adding all of these terms yields the expression 9x^{2} − 6x − 6x + 4. After combining like terms, the expression is 9x^{2} − 12x + 4.
What is the slope of the line given by the equation 3x + 4y = −12 ?

Solution
C. The line 3x + 4y = −12 is given in standard form. To find the slope, convert the equation into slopeintercept form (y = mx+b, where m is the slope and b is the yintercept):
(3x + 4y) − 3x = −12 − 3x
\(\frac{(4y)}{4}=\frac{(−3x − 12)}{4}\)
y =(^{3}⁄_{4})x − 3
Since the equation is now in slopeintercept form, it is easy to see that the slope is ^{3}⁄_{4}.