If a^{2} + b^{2} = z and ab = y, which of the following is equivalent to 4 +8 z y ?

Solution
Substituting a^{2} + b^{2} for z and ab for y into the expression 4z + 8y gives 4(a^{2} + b^{2}) + 8ab. Multiplying a^{2} + b^{2} by 4 gives 4a^{2} + 4b^{2} + 8ab, or equivalently 4(a^{2} + 2ab + b^{2}).Since (a^{2} + 2ab + b^{2}) = (a + b)^{2}, it follows that 4z + 8y is equivalent to (2a + 2b)^{2}.
Choices A, C, and D are incorrect and likely result from errors made when substituting or factoring.
y = x^{2}
2y + 6 = 2(x + 3)
If (x, y) is a solution of the system of equations above and x > 0, what is the value of xy ?

Solution
Substituting x^{2} for y in the second equation gives 2(x^{2}) + 6 = 2(x + 3). This equation can be solved as follows:
2x^{2} + 6 = 2x + 6 (Apply the distributive property.)
2x^{2} + 6 − 2x − 6 = 0 (Subtract 2x and 6 from both sides of the equation.)
2x^{2} − 2x = 0 (Combine like terms.)
2x(x − 1) = 0 (Factor both terms on the left side of the equation by 2x.)
Thus, x = 0 and x = 1 are the solutions to the system. Since x > 0, only x = 1 needs to be considered. The value of y when x = 1 is y = x^{2} = 1^{2} = 1. Therefore, the value of xy is (1)(1) = 1.
Choices B, C, and D are incorrect and likely result from a computational or conceptual error when solving this system of equations.
In air, the speed of sound S, in meters per second, is a linear function of the air temperature T, in degrees Celsius, and is given by ST(T) = 0.6T + 331.4. Which of the following statements is the best interpretation of the number 331.4 in this context?

Solution
The constant term 331.4 in S(T) = 0.6T + 331.4 is the value of S when T = 0. The value T = 0 corresponds to a temperature of 0°C. Since S(T) represents the speed of sound, 331.4 is the speed of sound, in meters per second, when the temperature is 0°C.
Choice B is incorrect. When T = 0.6°C, S(T) = 0.6(0.6) + 331.4 = 331.76, not 331.4, meters per second. Choice C is incorrect. Based on the given formula, the speed of sound increases by 0.6 meters per second for every increase of temperature by 1°C, as shown by the equation 0.6(T + 1) + 331.4 = (0.6T + 331.4) +0.6. Choice D is incorrect. An increase in the speed of sound, in meters per second, that corresponds to an increase of 0.6°C is 0.6(0.6) = 0.36.
Jackie has two summer jobs. She works as a tutor, which pays $12 per hour, and she works as a lifeguard, which pays $9.50 per hour. She can work no more than 20 hours per week, but she wants to earn at least $220 per week. Which of the following systems of inequalities represents this situation in terms of x and y, where x is the number of hours
she tutors and y is the number of hours she works as a lifeguard?

Solution
If Jackie works x hours as a tutor, which pays $12 per hour, she earns 12x dollars. If Jackie works y hours as a lifeguard, which pays $9.50 per hour, she earns 9.5y dollars. Thus the total, in dollars, Jackie earns in a week that she works x hours as a tutor and y hours as a lifeguard is 12x + 9.5y. Therefore, the condition that Jackie wants to earn at least $220 is represented by the inequality 12x + 9.5y ≥ 220. The condition that Jackie can work no more than 20 hours per week is represented by the inequality x + y ≤ 20. These two inequalities form the system shown in choice C.
Choice A is incorrect. This system represents the conditions that Jackie earns no more than $220 and works at least 20 hours. Choice B is incorrect. The first inequality in this system represents the condition that Jackie earns no more than $220. Choice D is incorrect. The second inequality in this system represents the condition that Jackie works at least 20 hours.
Which of the following is equivalent to the sum of the expressions a^{2} − 1 and a + 1 ?

Solution
The sum of (a^{2} − 1) and (a + 1) can be rewritten as (a^{2} − 1) + (a + 1), or a^{2} − 1 + a + 1, which is equal to a^{2} + a + 0. Therefore, the sum of the two expressions is equal to a^{2} + a.
Choices B and D are incorrect. Since neither of the two expressions has a term with a^{3}, the sum of the two expressions cannot have the term a^{3} when simplified. Choice C is incorrect. This choice may result from mistakenly adding the terms a^{2} and a to get 2a^{2}.
In the equation above, k is a constant. If x = 9, what is the value of k ?

Solution
If x = 9 in the equation \(\sqrt{K+2}x=0\), this equation becomes \(\sqrt{K+2}9=0\),which can be rewritten as \(\sqrt{K+2}=9\). Squaring each side of \(\sqrt{K+2}=9\) gives k + 2 = 81, or k = 79. Substituting k = 79 into the equation \(\sqrt{K+2}9=0\) confirms this is the correct value for k.
Choices A, B, and C are incorrect because substituting any of these values for k in the equation \(\sqrt{K+2}9=0\) gives a false statement. For example, if k = 7, the equation becomes \(\sqrt{K+2}9=\sqrt{9}9=39=0\) which is false.
Which of the following is an example of a function whose graph in the xyplane has no xintercepts?

Solution
If f is a function of x, then the graph of f in the xyplane consists of all points (x, f(x)). An xintercept is where the graph intersects the xaxis; since all points on the xaxis have ycoordinate 0, the graph of f will cross the xaxis at values of x such that f(x) = 0. Therefore, the graph of a function f will have no xintercepts if and only if f has no real zeros. Likewise, the graph of a quadratic function with no real zeros will have no xintercepts.
Choice A is incorrect. The graph of a linear function in the xyplane whose rate of change is not zero is a line with a nonzero slope. The xaxis is a horizontal line and thus has slope 0, so the vgraph of the linear function whose rate of change is not zero is a line that is not parallel to the xaxis. Thus, the graph must intersect the xaxis at some point, and this point is an xintercept of the graph. Choices B and D are incorrect because the graph of any function with a real zero must have an xintercept.
What are the solutions of the quadratic equation 4x^{2} − 8x − 12 = 0?

Solution
Dividing both sides of the quadratic equation 4x^{2} − 8x − 12 = 0 by 4 yields x^{2} − 2x − 3 = 0. The equation x^{2}− 2x − 3 = 0 can be factored as (x + 1)(x − 3) = 0. This equation is true when x + 1 = 0 or x − 3 = 0. Solving for x gives the solutions to the original quadratic equation: x = −1 and x = 3.
Choices A and C are incorrect because −3 is not a solution of 4x^{2} − 8x − 12 = 0: 4(−3)^{2} − 8(−3) −12 = 36 + 24 − 12 ≠ 0. Choice D is incorrect because 1 is not a solution of 4x^{2} − 8x − 12 = 0: 4(1)^{2} − 8(1) − 12 = 4 − 8 − 12 ≠ 0.
The circle above with center O has a circumference of 36. What is the length of minor arc \(\overset{\frown}{AC}\)?

Solution
A circle has 360 degrees of arc. In the circle shown, O is the center of the circle and angle AOC is a central angle of the circle. From the figure, the two diameters that meet to form angle AOC are perpendicular, so the measure of angle AOC is 90°. This central angle intercepts minor arc AC, meaning minor arc AC has 90° of arc. Since the circumference (length) of the entire circle is 36, the length of minor arc AC is \(\frac{90}{360}\times 36=9\).
Choices B, C, and D are incorrect. The perpendicular diameters divide the circumference of the circle into four equal arcs; therefore, minor arc AC is ^{1}⁄_{4} of the circumference. However, the lengths in choices B and C are, respectively,^{1}⁄_{3} and ^{1}⁄_{2} the circumference of the circle, and the length in choice D is the length of the entire circumference. None of these lengths is ^{1}⁄_{4} the circumference.
Which of the following is an equation of line A in the xyplane above?

Solution
From the graph, the yintercept of line is (0, 1). The line also passes through the point (1, 2). Therefore the slope of the line is \(\frac{21}{10}\) = ^{1}⁄_{1} = ^{1}⁄_{1} = 1, and in slopeintercept form, the equation for line l is y = x + 1.
Choice A is incorrect. It is the equation of the vertical line that passes through the point (1, 0). Choice B is incorrect. It is the equation of the horizontal line that passes through the point (0, 1). Choice C is incorrect. The line defined by this equation has yintercept (0, 0), whereas line l has yintercept (0, 1).