ANALISIS DE ALTERNATIVAS
5.05 Análisis e Interpretación De Datos
If a2 +14a= 51= and a > 0, what is the value of a + 7? content: Passport to Advanced Math
key: 10
calculator usage: No calculator
There is more than one way to solve this problem. A student can apply standard techniques by rewriting the equation a2 +14a=51 as
a2 +14a−51 0= and then factoring. Since the coefficient of a is 14 and the constant term is –51, factoring a2 +14a−51 0= requires writing 51 as the product of two numbers that differ by 14. This is 51=
( )( )
3 17 ,which gives the factorization a2 +14a−51=
(
a+17)(
a−3)
=0. The possible values of a are a = –17 and a = 3. Since it is given that a > 0, it must be true that a = 3. Thus, the value of a + 7 is 3 + 7 = 10.A student could also use the quadratic formula to find the possible values of a:
a = −14 ± 14
( )
−4 1 51( )(
−)
= − ± − −(
)
= − ± = − ± 2 1 14 196 204 2 14 2 400 14 202 2 .The possible values of a are a = − −14 202 = −17 and a= − +14 202 =3.
Again, since it is given that a > 0, it must be true that a = 3. Thus, the value of a + 7 is 3 + 7 = 10.
There is another way to solve this problem that will reward the student who recognizes that adding 49 to both sides of the equation yields
a2 +14a+49 51 49= + , or rather (a+7)2 =100, which has a perfect square on each side. Since a > 0, the solution a + 7 = 10 is evident.
SAMPLE 18
The function f is defined by f x( ) =2x3 +3x2 +cx+8, where c is a constant. In the xy-plane, the graph of f intersects the x-axis at the three points
(
−4 0, ,)
( )
12, ,0 and( )
p, .0 What is the value of c ?A) –18 B) –2 C) 2 D) 10
content: Passport to Advanced Math key: A
calculator usage: Calculator
Students could tackle this problem in many different ways, but the focus is on their understanding of the zeros of a polynomial function and how they are used to construct algebraic representations of polynomials. Choice A is correct. The given zeros can be used to set up an equation to solve for c. Substituting –4 for x and 0 for y yields –4c = 72, or c = –18. Alternatively, since –4, 12, and p are zeros of the polynomial function
f x( ) =2x3 +3x2 +cx+8, it follows that f x( ) (= 2x−1)(x +4)(x p− ).
Were this polynomial multiplied out, the constant term would be
( )( )( )−1 4 − =p 4p. (We can see this without performing the full expansion.) Since it is given that this value is 8, it goes that 4p = 8 or rather, p = 2. Substituting 2 for p in the polynomial function yields
f x( ) (= 2x−1)(x +4)(x−2), and after multiplying the factors one finds that the coefficient of the x term, or the value of c, is –18.
Choice B is not the correct answer. This value is a misunderstood version of the value of p, not c, and the relationship between the zero and the factor (if a is the zero of a polynomial, its corresponding factor is x – a) has been confused.
Choice C is not the correct answer. This is the value of p, not c. Using this value as the third factor of the polynomial will reveal that the value of c is –18. Choice D is not the correct answer. This represents a sign error in the final step in determining the value of c.
SAMPLE 19 x y y x x y 2 2 2 5 3 1 + = = − − =
A system of three equations and their graphs in the xy-plane are shown above. How many solutions does the system have?
A) One B) Two C) Three D) Four
content: Passport to Advanced Math key: B
calculator usage: Calculator
Students must also demonstrate their conceptual understanding of mathematics. In this problem, the word “solutions” needs to be contextualized within the situation described, and students must recognize that only points where all three graphs have a point in common are solutions to the system.
Choice B is correct. The solutions to the system of equations are the points where the circle, parabola, and line all intersect. These points are (–1, 2) and (2, 1), and these are the only solutions to the system.
Choice A is not the correct answer. This answer may reflect the misconception that a system of equations can have only one solution. Choice C is not the correct answer. This answer may reflect the misconception that a system of equations has as many solutions as the number of equations in the system.
x y 1 2 3 –1 –3 –2 –1 –2 –3 1 2 3 0
Choice D is not the correct answer. This answer may reflect the misconception that the solutions of the system are represented by the points where any two of the curves intersect, rather than the correct concept that the solutions are represented only by the points where all three curves intersect.
SAMPLE 20
What is one possible solution to the equation x24+1− x12−1=1?
content: Passport to Advanced Math key: 5 or 7
calculator usage: No calculator
Students should look for the best solution methods for solving rational equations before they begin. Looking for structure and common denominators will prove very useful at the onset, and will help prevent complex computations that do not lead to a solution.
In this problem, multiplying both sides of the equation by the common denominator (x+1)(x−1)yields 24(x− −1 12) (x + =1) (x+1)(x−1).
Multiplication and simplification then yields 12x−36 =x2 −1,or
x2 −12x +35 0= . Factoring the quadratic gives (x −5)(x −7) =0,so the solutions occur at x = 5 and x = 7, both of which should be checked in the original equation to ensure that they are not extraneous. In this case, both values are solutions.
SAMPLE 21
1 2 1
5
x + x =
Anise needs to complete a printing job using both of the printers in her office. One of the printers is twice as fast as the other, and together the printers can complete the job in 5 hours. The equation above represents the situation described. Which of the following describes what the expression
1
x represents in this equation?
A) The time, in hours, that it takes the slower printer to complete the printing job alone
B) The portion of the job that the slower printer would complete in one hour
C) The portion of the job that the faster printer would complete in two hours
D) The time, in hours, that it takes the slower printer to complete 15 of the printing job
content: Passport to Advanced Math key: B
calculator usage: No calculator
This question requires students to interpret/decipher an expression/ equation (already constructed for them) that models a real-world situation. This is an important skill to develop and a good practice to foster students’ reasoning skills. The students who answer this question correctly are able to interpret the whole expression (or specific parts) in terms of its context.
Choice B is correct. From the description given, 15 is the portion of the job that the two printers, working together, can complete in one hour, and each term in the sum on the left side is the part of this 15 of the job that one of the printers contributes. Since one of the printers is twice as fast as the other, 2x describes the portion of the job that the faster printer is able to complete in one hour and 1x describes the portion of the job that the slower printer is able to complete in one hour.
Choice A is not the correct answer. The student may have not seen that in this context, the rates (that is, the work completed in a fixed time) of
the printers can be added to get the combined rate, but the times it takes each printer to complete the job cannot be added to get the time for both printers working together, since the time for printers working together is less than, not greater than, the times for each printer alone. Hence the terms in the sum cannot refer to hours worked. In fact, the time it would take the slower printer to complete the whole job is x hours.
Choice C is not the correct answer. The student may have seen that
1
x is the smaller term in the sum, wrongly concluded that the smaller
term must apply to the faster printer, and then assumed the 2 in the numerator of the second term implies the equation describes work completed in 2 hours. In fact, the portion of the job that the faster printer could complete in 2 hours is
( )
2 2( )
x = 4x.Choice D is not the correct answer. The student may have correctly seen that the value 15 on the right side refers to the portion of the job completed, but not seen that in this context, the rates (that is, the work completed in a fixed time) of the printers can be added to get the combined rate, but the times it takes each printer to complete the job cannot be added to get the time for both printers working together. Hence the terms in the sum cannot refer to hours worked. In fact, the time it takes the slower printer to complete 15 of the job is x5 hours.
SAMPLE 22 x y y x 2 2 153 4 + = = −
If (x, y) is a solution to the system of equations above, what is the value of x2? A) –51
B) 3 C) 9 D) 144
content: Passport to Advanced Math key: C
calculator usage: Calculator
In this question, students are asked to manipulate one equation to use in another in order to solve for a given value, making use of structure where appropriate.
Choice C is correct. The second equation gives y in terms of x, so a student can use this to rewrite the first equation in terms of x. Substituting –4x for y in the equation x2 +y2 =153 gives
x2 + −
(
4x)
2 =153. This can be simplified to x2 +16x2 =153, or17x2 =153. Since the question asks for the value of x2, not x, dividing both sides of 17x2 =153 by 17 gives the answer: x2 = 15317 = 9.
Choice A is not the correct answer. This answer may result from neglecting to square the coefficient –4 in y = –4x, which would give y2 = −4x2. Then the first equation would become
x2 −4x2 = −3x2 = 153, which would give –51 as the value of x2.
Choice B is not the correct answer. This answer may result from finding the value for x,not the value of x2.
Choice D is not the correct answer. This answer may result from finding the value of y2, not x2.
SAMPLE 23
If the expression 24xx−21 is written in the equivalent form 2x1−1 + A, what is A in terms of x?
A) 2x + 1 B) 2x – 1 C) 4x2 D) 4x2 − 1
content: Passport to Advanced Math key: A
calculator usage: Calculator
This question assesses the ability to transform a given expression in a more useful form (from improper to proper rational form). There are multiple approaches to this problem, some longer and more routine, some faster and more insightful.
Choice A is correct. The form of the equation suggests performing long division on 24xx−21: 2 1 4 4 2 2 2 1 2 2 x x x x x x − − −
)
11 2x+1Since the remainder 1 matches the numerator in 1
2x −1,it is clear that
A =2x+1.
A short way to find the answer is to use the structure to rewrite the numerator of the expression as (4x2 − +1 1) , recognizing the term in parentheses as a difference of squares, making the expression equal to
(2 1 2)( 1 1)
2 1 2 1 2 1 1
x x
x x x
− + +
− = + + − . From this, the answer 2x + 1
is apparent. Another way to find the answer is to isolate A in the form
A = 4xx− = x−
2 1 2 1 1
2
and simplify. As with the first approach, this approach also requires students to recognize 4x2 −1as a difference of squares that factors.
Choice B is not the correct answer. The student may have made a sign error while subtracting partial quotients in the long division.
Choice C is not the correct answer. The student may misunderstand how to work with fractions and may have tried the incorrect calculation
4 2 1 1 4 2 1 2 1 1 4 2 2 2 x x x x x x − =
( )( )
− = − + .Choice D is not the correct answer. The student may misunderstand how to work with fractions and may have tried the incorrect calculation
4
2 1 1 42 1 1 2 1 1 4 1
2 2 2
x