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Answer Key for California State Standards: Algebra I

19.0: Students know the quadratic formula and are familiar with its proof by
completing the square.

a. Given a quadratic equation of the form: ax2 + bx + c = 0, a ≠ 0
What is the formula for finding the solutions to the equation ?

b. The equations below are part of a derivation of the quadratic
formula by completing the square:

Which of the following is the best next step for the
derivation of the quadratic formula?

The derivation of the quadratic formula by completing the square is
standard
material in any good Algebra I textbook.

 

20.0: Students use the quadratic formula to find the roots of a second-degree
polynomial and to solve quadratic equations.

Find all values of x which satisfy the equation 4x2 - 4x - 1 = 0

The roots are and

 

21.0: Students graph quadratic functions and know that their roots are the x-intercepts.

You may assume that the following equation is correct for all values
of x:

a. For which values of x, if any, does the graph of the equation
cross the x axis?

The graph of crosses the x-axis at and

b. Sketch the graph of the equation

 

22.0: Students use the quadratic formula or factoring techniques or both to determine
whether the graph of quadratic function will intersect the x-axis in zero, one or two points.

Use the quadratic formula or the method of factoring to determine whether the
graphs of the following functions intersect the x axis in zero, one, or two points.
(Do not graph the functions.)

Each of these problems may be solved using the discriminant, D = b2 - 4ac, which
appears under the radical sign in the quadratic formula. If D > 0, the graph has
exactly two x-intercepts. If D = 0, the graph has exactly one
x-intercept. If D < 0, the graph does not intersect the x-axis.

a. D = b2 - 4ac = 12 - 4 · 1 · 1 < 0

Therefore the graph of y = x2 + x + 1 does not intersect the x-axis
( equivalently x 2 + x + 1 = 0 has no real solutions).

Answer: 0

b. D = b2 - 4ac = 122 - 4 · 4 · 5 > 0. Therefore y = 4x2 + 12x + 5 has two
x-intercepts. This may also be seen by factoring:

4x2 + 12x + 5 = (2x + 5)(2x + 1)

So the intercepts are and .

Answer: 2

c. D = b2 - 4ac = (-12)2 - 4 · 9 · 4 = 122 - (3 · 4)(3 · 4) = 122 - 122 = 0. Therefore y
= 9x2 - 12x + 4 has exactly one x-intercept. This may also be seen by factoring:

9x2 - 12x + 4 = (3x - 2)2
Setting this expression equal to zero gives exactly one solution, .

Answer: 1

 

23.0: Students apply quadratic equations to physical problems, such as the motion of an
object under the force of gravity.

a. If an object is thrown vertically with an initial velocity of from
an initial height of feet, then neglecting air friction its
height h(t) in feet above the ground t seconds after the ball was
thrown is given by the formula

If a ball is thrown upward from the top of a 144 foot tower at 96 feet per
second, how long will it take for the ball to reach the ground if there is no air
friction and the path of the ball is unimpeded?

Let h(t) be the height above the ground at time t measured in seconds. Then

h(t) - 16t2 + 96t + 144

In order to find t such that h(t) is zero, set h(t) = 0 and solve for t.

Since the object was thrown at t = 0 and time moves forward, the correct
solution is seconds.

b. The boiling point of water depends on air pressure and air pressure
decreases with altitude. Suppose that the height H above the
ground in meters can be deduced from the temperature T at which
water boils in degress Celsius by the following formula:

H = 1000(100 - T) + 580(100 - T)2

1. If water on the top of a mountain boils at 99.5 degrees Celsius,
how high is the mountain?

2. What is the approximate boiling point of water at sea-level (H=0 meters)
according to this equation? Round your answers to the nearest 10 degrees.

The temperature at which water boils at sea level according to the formula
may be deduced by setting H = 0 and solving for T:

The equation predicts the boiling point is approximately 100°C

 

24.0: Students use and know simple aspects of a logical argument:
24.1: Students explain the difference between inductive and deductive reasoning and provide
examples of each.
24.2: Students identify the hypothesis and conclusion in a logical deducation.
24.3: Students use counterexamples to show that an assertion is false and recognize that a single
counterexample is sufficient to refute an assertion.

a. Verify to your own satisfaction, by direct calculation, the
correctness of the following equations (do not submit your
calculations on this exam):

1. Using inductive reasoning, propose a formula that gives the sum
for for any counting number n .

2. Does the sequence of formulas above prove that your answer to part 1 is
correct? Explain your answer.

No, inductive reasoning is really a form of guessing based on previous
observations.
[Note to the reader: In this case the formula given in part 1 is correct
for any value of n. Inductive reasoning worked in this case, but it
doesn't always give correct answers.]

b. Consider the following mathematical statement :

If y is a positive integer, then 1 + 1141y2 is not a perfect square.

1. Write the hypothesis of this statement.

y is a positive integer

2. Write the conclusion of this statement.
 

1 + 1141y2 is not a perfect square


3. Use whole number arithmetic to prove that the conclusion is
correct when y = 1.

If y = 1, 1 + 1141y2 = 1142. To show that 1142 is not a perfect
square, it suffices to show that 1142 falls between the
squares of two consecutive integers.

332 = 1089 < 1142 < 1156 = 342

Therefore the conclusion, 1 + 1141y2 is not a perfect square, is
correct when y = 1.

4. It has been shown by mathematicians that the conclusion is correct for
each positive integer y up to and including 30,693,385,322,765,657,197,397,207.
However, if this number is increased by 1 so that

y = 30,693,385,322,765,657,197,397,208

then the positive square root of 1 + 1141y2 is

1,036,782,394,157,223,963,237,125,215

Is the statement, "If y is a positive integer, then 1 + 1141y2
is not a perfect square" correct? Explain your answer.

No, the statement is incorrect because the conclusion is
false for at least one positive integer value of y.
[Note that inductive reasoning for this problem would
most likely lead to a faulty conclusion.]

 

25.0: Students use properties of the number system to judge the validity of results,
to justify each step of a procedure, and to prove or disprove statements:
25.1: Students use properties of numbers to construct simple, valid arguments (direct and
indirect) for, or formulate counterexamples to, claimed assertions.
25.2: Students judge the validity of an argument according to whether the properties of the
real number system and the order of operations have been applied correctly at each step.
25.3: Given a specific algebraic statement involving linear, quadratic, or absolute value
expressions or equations or inequalities , students determine whether the statement is true
sometimes, always, or never.

a. Prove, using basic properties of algebra, or disprove by finding a
counterexample, each of the following statements:

1. The set of even numbers is closed under addition .

A number m is even if and only if m = 2k for some integer k. Let m and n
be even and let m = 2k and m = 2j for integers k and j. Then:

m + n = 2k + 2j = 2(k + j)

Therefore m + n has a factor of 2 so it is even. This proves that the sum
of any two even numbers is even and therefore the set of even numbers
is closed under addition.

2. The sum of any two odd numbers is even.

A number m is odd if and only if m = 2k + 1 for some integer k. Let m and n
be odd and let m = 2k +1 and n = 2j + 1 for integers k and j. Then:

m + n = (2k + 1) + (2j + 1) = 2k + 2j + 2 = 2(k + j + 1)

Therefore m + n has a factor of 2 so it is even. This proves that the sum
of any two odd numbers is odd.

a.

3. For any positive real number

This statement is false. gives a counterexample because:

b. Find all possible pairs of numbers a and b which satisfy the
equation a2 + b2 = (a + b)2. Explain your reasoning.

Suppose a2 + b2 = (a + b)2
Then a2 + b2 = a2 + 2ab + b2
Therefore 0 = 2ab
Therefore ab = 0
Therefore a = 0 or b = 0.
If a = 0 or b = 0 then a2 + b2 = (a + b)2.
We conclude that a2 + b2 = (a + b)2 if and only if a = 0 or b = 0

c. Identify the step below in which a fallacy occurs:

Step 1: Let a = b = 1
Step 2: a2 = ab
Step 3: a2 - b2 = ab - b2
Step 4: (a - b)(a + b) = b(a - b)
Step 5: a + b = b
Step 6: 2 = 1

Answer: Step 5

Explain why the step you have chosen as the fallacy is incorrect

Step 5 results from dividing both sides of the previous equation by
zero because a - b = 0.

d. Is the following equation true for some values of x, no values of x or all values
of x?

The equation is true for all values of x because for any
value of x:

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