1. A---- equation can be written in the form ax^2+bx+c=0 where a,b and c are real numbers, and ais a nonzero number
Linear
Quadratic
Cubic
Quartic
2. The number of vinyl album sales
y (in millions) in a country x years after 2010 can be modelled for 2010 through 2016 with the function below. y=0.12x^2+0.3x+3.3 Use this model to predict the number of vinyl album sales in the country in the year 2020. (x=10) (x=10)
A. The number of vinyl album sales in the country in the year 2020 will be 18.3 million.
B. The number of vinyl album sales in the country in the year 2020 will be 21.12 million.
C. The number of vinyl album sales in the country in the year 2020 will be 20.5 million.
D. The number of vinyl album sales in the country in the year 2020 will be 7.74 million.
3. Josephine solved a quadratic equation:(x+6)^2=49
Step 1 squared (x+6)^2= squared 49
Step 2 x+6=7
Step 3 7−6x=7−6
Step 4 x=1
Did Josephine make a mistake?
A. Yes, Josephine forgot to consider the negative square root of 49, which is -7, which leads to two equations to solve.
B. No, Josephine made no mistakes in solving this equation.
C. Yes, Josephine should have subtracted 6 from each side to start with to get x^2=43
D. Yes, Josephine should have multiplied out 2(x+6)^2 first
4. What are the solutions to the equation (m+1)^2 +1=5
A. M=-1
B. M=-3
C. m=1
D. m=3
E. m=-5
F. m=5
5. Which of the following rewrites shows the correct process for completing the square?
A. X^2+8x = 15 rewritten as x^2+8+16=15+16
B. 4x^2+10x=−5 rewritten as 4x^2+10x=−5=25
C. 4x^2+10x=−5 rewritten as 4x^2+10x+100=-5+100
D. X^2+8x=15 rewritten as X^2+8x +64=15+64
6. . Jesse and Benjamin are asked to solve 100x^2− 100=0 Here are their solutions. Who is correct? Explain.
Jesse
100x^2−100=0
(10x)^2−10^2=0
(10x−10)(10x+10)=0
10(x−1)⋅10(x+1)=0
(x−1)=0 or (x+1)=0
X=1 or x=-1
Benjamin
100x^2−100
=100(x^2−1)
= 100(x−1)(x+1)
(x−1)=0 or (x−1)=0
X=1or -1
A. Neither of them are correct. They both used a difference of perfect squares pattern which does not apply here.
B. Benjamin is correct they solved it in less steps.
C. Jesse is correct they showed more work.
D. Both are correct, they both correctly applied mathematical properties and arrived at the same solution.
7. Create a quadratic inequality that represents the following phrase: The square of the quantity 5 subtracted from x has a minimum value of 50
A. (x−5)^2≥50
B. (x−5)^2>50
C. (x+5)2>50
D. (x+5)2≥50
1. Quadratic
2. C. The number of vinyl album sales in the country in the year 2020 will be 20.5 million.
3. A. Yes, Josephine forgot to consider the negative square root of 49, which is -7, which leads to two equations to solve.
4. B. M=-3
5. D. X^2+8x=15 rewritten as X^2+8x +64=15+64
6. D. Both are correct, they both correctly applied mathematical properties and arrived at the same solution.
7. B. (x−5)^2>50
The profit of a company producing
X thousand units is given by p (x)=2x^2+38x-120 Find the values of x for which the company makes a profit. Explain and justify your answer.
Hint: The company makes a profit when
P(x)>0]
To find the values of x for which the company makes a profit, we need to determine when the profit function P(x) is greater than 0.
The profit function is given by P(x) = 2x^2 + 38x - 120.
To determine when P(x) > 0, we need to find the roots of the quadratic equation 2x^2 + 38x - 120 = 0, since the graph of a quadratic function intersects the x-axis at its roots.
To solve the quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula.
The quadratic formula states that for any quadratic equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a).
In our case, a = 2, b = 38, and c = -120.
Using the quadratic formula, we have:
x = (-38 ± √(38^2 - 4(2)(-120))) / (2(2))
x = (-38 ± √(1444 + 960)) / 4
x = (-38 ± √(2404)) / 4
x = (-38 ± 49) / 4
Thus, we have two potential values for x:
1) x = (-38 + 49) / 4 = 11 / 4 = 2.75
2) x = (-38 - 49) / 4 = -87 / 4 = -21.75
Since the profit function represents the profit of the company producing x thousand units, the values of x must be positive. Therefore, the company makes a profit when x = 2.75 thousand units.
1. Quadratic
2. The number of vinyl album sales in the country in the year 2020 will be 20.5 million. (Option C)
3. Yes, Josephine made a mistake. (Option A)
4. The solutions to the equation are m = -3 and m = 1. (Options B and C)
5. The correct rewrite is option D: X^2+8x=15 rewritten as X^2+8x+64=15+64
6. Both Jesse and Benjamin are correct. (Option D)
7. The quadratic inequality that represents the given phrase is (x-5)^2 > 50. (Option B)
1. The equation can be written in the form ax^2+bx+c=0 where a, b, and c are real numbers, and a is a nonzero number. This is known as a quadratic equation.
2. To predict the number of vinyl album sales in the country in the year 2020, we can substitute x=10 into the given function y=0.12x^2+0.3x+3.3.
Plugging in x=10, we get y = 0.12(10)^2 + 0.3(10) + 3.3 = 12 + 3 + 3.3 = 18.3
Therefore, the correct answer is A. The number of vinyl album sales in the country in the year 2020 will be 18.3 million.
3. Let's go through Josephine's steps to see if there are any mistakes.
Step 1: (x+6)^2 = 49
Step 2: x+6 = 7
Step 3: 7-6x = 7-6
Step 4: x = 1
Josephine correctly solved the quadratic equation. The original equation has one solution: x = 1.
Therefore, the correct answer is B. No, Josephine made no mistakes in solving this equation.
4. To find the solutions to the equation (m+1)^2 + 1 = 5, we need to simplify and solve for m.
(m+1)^2 + 1 = 5
m^2 + 2m + 1 + 1 = 5
m^2 + 2m + 2 = 5
m^2 + 2m - 3 = 0
Now we have a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 2, and c = -3. We can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Factors of -3 that add up to 2 are 3 and -1. Therefore, we can factor the quadratic equation as (m+3)(m-1) = 0.
Setting each factor equal to zero gives us two possible solutions: m+3=0 or m-1=0. So, m = -3 or m = 1.
Therefore, the solutions to the equation are m = -3 and m = 1. The correct answers are A. m = -1 and C. m = 1.
5. To complete the square for the equation x^2+8x=15, we need to follow these steps.
Step 1: Move the constant term (15) to the other side of the equation: x^2 + 8x - 15 = 0
Step 2: Take half of the coefficient of x (8) and square it: (8/2)^2 = 16
Step 3: Add the squared term (16) to both sides of the equation: x^2 + 8x + 16 - 15 = 16
Step 4: Factor the perfect square trinomial on the left side and simplify: (x + 4)^2 = 16 + 15
Step 5: Simplify further: (x + 4)^2 = 31
Therefore, the correct answer is D. x^2 + 8x = 15 rewritten as x^2 + 8x + 64 = 15 + 64.
6. Let's examine Jesse and Benjamin's solutions to the equation 100x^2 - 100 = 0.
Jesse:
100x^2 - 100 = 0
(10x)^2 - 10^2 = 0
(10x - 10)(10x + 10) = 0
10(x - 1) * 10(x + 1) = 0
(x - 1) = 0 or (x + 1) = 0
x = 1 or x = -1
Benjamin:
100x^2 - 100
= 100(x^2 - 1)
= 100(x - 1)(x + 1)
(x - 1) = 0 or (x - 1) = 0
x = 1 or x = -1
Both Jesse and Benjamin arrived at the same solutions: x = 1 and x = -1. So, the correct answer is D. Both are correct, they both correctly applied mathematical properties and arrived at the same solution.
7. The phrase "The square of the quantity 5 subtracted from x has a minimum value of 50" can be represented as (x - 5)^2 ≥ 50.
Therefore, the correct answer is A. (x - 5)^2 ≥ 50.