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]
but make it look like a 10th greder did it
To determine the values of x for which the company makes a profit, we need to find the values of x that satisfy the inequality P(x) > 0.
Given that the profit function is P(x) = 2x^2 + 38x - 120, we need to find the values of x that make this expression greater than zero.
To simplify, we can factor the quadratic equation:
P(x) = (x+10)(2x-12)
Now, we can set each factor equal to zero and solve for x:
x + 10 = 0
x = -10
2x - 12 = 0
2x = 12
x = 6
Therefore, the company makes a profit when x is greater than 6 or less than -10.
To justify this answer, we can analyze the factors of the profit function:
1. (x+10): When x > -10, this factor is positive, contributing to a positive profit.
2. (2x-12): When x > 6, this factor is positive, contributing to a positive profit.
Therefore, the company makes a profit when x > 6 or x < -10, as these values satisfy the condition P(x) > 0.
1. In the complex number a+bi,a is called the ---- and b is called the---
Real part
Imaginary part
2. The result of adding a complex number to its conjugate is----
The result of subtracting a complex number to its conjugate is---
A real number
A pure Imaginary number
A complex number
3. Sarah gave the following justification that
(x−i)^2=x^2−2ix+1is an identity. Is Sarah correct?
step (1x−i)^2=(x−i)(x−i)
step 2 (x−i)(x−i)=x^2−ix−ix−i^2
step3 x^2−ix−ix−i^2=x^2−2ix−i^2
step4 x^2−2ix−i^2=x^2−2ix+1
A. Sarah is correct.
B. Sarah is incorrect. She made a mistake in step 2.
C. Sarah is incorrect. She made a mistake in step 3.
D. Sarah is incorrect. She made a mistake in step 4.
1. In the complex number a+bi, a is called the real part and b is called the imaginary part.
2. The result of adding a complex number to its conjugate is a real number.
The result of subtracting a complex number to its conjugate is also a real number.
3. Sarah is incorrect. She made a mistake in step 3.
In step 3, it should be x^2 - ix - ix + i^2 instead of x^2 - ix - ix - i^2.
Therefore, the correct option is D. Sarah is incorrect. She made a mistake in step 4.
The discriminant of a quadratic equation is given by
B^2−4ac
Explain in your own words why a quadratic equation has...
...two real solutions if the discriminant is positive
...one real solution if the discriminant is zero
...no real solutions if the discriminant is negative
[Hint: think about the quadratic formula]
But make it look like a 10th-grader did it
Hey! So when we talk about the discriminant of a quadratic equation, we're basically looking at the stuff inside the square root in the quadratic formula, which is B^2−4ac. Let's break it down and see why it affects the number of solutions a quadratic equation has.
First off, if the discriminant is positive, it means that the expression B^2−4ac is greater than zero. According to the quadratic formula, which is x = (-B ± √(B^2−4ac))/(2a), the term inside the square root will give us a positive value. When we add or subtract a positive number from -B, we'll get two different values for x, which means there are two real solutions!
Now, if the discriminant is zero, it means that the expression B^2−4ac equals zero. In the quadratic formula, when the square root is zero, it doesn't really change the value of x. So, since the ± term doesn't have any effect, we end up with only one value for x. This means there's just one real solution!
Finally, when the discriminant is negative, it means that the expression B^2−4ac is less than zero. Here's where things get a bit tricky. In the quadratic formula, when we have a negative value inside the square root, we can't work with that in real numbers. You see, because the square root of a negative number gives us an imaginary number, it means that there are no real solutions! It's kind of like saying we can't find a real value for x that would make the equation work.
So, to sum it up:
- A positive discriminant gives us two real solutions.
- A zero discriminant gives us one real solution.
- A negative discriminant gives us no real solutions.
And all this comes from understanding what the discriminant tells us about the quadratic equation through the quadratic formula. Pretty cool, right?
To determine the values of x for which the company makes a profit, we need to analyze the profit function, p(x), and identify the x-values that result in a positive profit.
Given: p(x) = 2x^2 + 38x - 120
To find the values of x for which the company makes a profit, we need to solve the inequality p(x) > 0, which represents the condition for positive profit.
Step 1: Write the inequality:
2x^2 + 38x - 120 > 0
Step 2: Solve the inequality:
To solve this quadratic inequality, we can use different methods such as factoring, completing the square, or using the quadratic formula. Since this quadratic expression is not easily factorable, we will use the quadratic formula.
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions can be found using the formula x = (-b ± √(b^2 - 4ac)) / 2a.
For our inequality, a = 2, b = 38, c = -120, and we want the solutions where p(x) is greater than zero.
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
x = (-(38) ± √(38^2 - 4(2)(-120))) / 2(2)
x = (-38 ± √(1444 + 960)) / 4
x = (-38 ± √2404) / 4
x = (-38 ± 49.03) / 4
Now we have two potential solutions for x:
x = (-38 + 49.03) / 4
x = 11.03 / 4
x ≈ 2.758
x = (-38 - 49.03) / 4
x = -87.03 / 4
x ≈ -21.757
Step 3: Analyze the solutions:
Since the company produces X thousand units, the number of units produced cannot be negative. Therefore, we must only consider the positive solution x ≈ 2.758.
Thus, the company makes a profit when it produces more than approximately 2,758 thousand units.
To determine the values of x for which the company makes a profit, we need to find the values that make P(x) greater than zero, where P(x) is the profit function defined by p(x) = 2x^2 + 38x - 120.
Let's break down the process step by step, making it easier to understand:
Step 1: Write down the profit function:
p(x) = 2x^2 + 38x - 120
Step 2: Set up the inequality for profit:
p(x) > 0
Step 3: Solve the inequality:
2x^2 + 38x - 120 > 0
Now, we will solve this quadratic inequality using the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using:
x = (-b ± √(b^2 - 4ac))/(2a)
In our case, the quadratic inequality is greater than zero, so we are looking for the values of x that make the expression positive.
Step 4: Apply the quadratic formula:
x = (-38 ± √(38^2 - 4*2*(-120))) / (2*2)
Simplifying further,
x = (-38 ± √(1444 + 960)) / 4
x = (-38 ± √(2404)) / 4
x = (-38 ± √(2^2 * 601)) / 4
x = (-38 ± 2√(601)) / 4
x = -9.5 ± 0.5√(601)
Now, we have two possible values for x: -9.5 + 0.5√(601) and -9.5 - 0.5√(601).
Therefore, the values of x for which the company makes a profit are approximately:
x ≈ -9.5 + 0.5√(601)
x ≈ -9.5 - 0.5√(601)
By substituting these approximate values back into the profit function, you can verify that the profit is indeed greater than zero for these values.
Note: The steps provided here are more detailed and follow a mathematical approach that may not be typically used by a 10th grader. However, understanding the concept and reasoning behind it is important for further mathematical development.