which constant term would mean that the expression is completely factored x ^ 2 - 3x +
-10
0
10
To completely factorize [tex]\(x^2 - 3x + c\),[/tex]the correct constant term is 10. With this, the expression becomes [tex]\((x - 2)(x - 5)\),[/tex] achieving complete factorization.
To find the correct constant term that would allow for complete factorization of the quadratic expression [tex]\(x^2 - 3x + c\),[/tex] let's consider what it means to factorize a quadratic expression:
A quadratic expression can be factored if it can be represented in the form [tex]\((x - a)(x - b)\),[/tex] where a and b are the roots of the expression. Given the original expression [tex]\(x^2 - 3x + c\),[/tex] we can expand [tex]\((x - a)(x - b)\)[/tex] and then compare coefficients to determine the constant term c.
Expanding [tex]\((x - a)(x - b)\)[/tex] :
- [tex]\((x - a)(x - b) = x^2 - (a + b)x + a \cdot b\).[/tex]
Comparing Coefficients :
- By comparing with [tex]\(x^2 - 3x + c\),[/tex] we can identify that [tex]\(a + b = 3\)[/tex] (the coefficient for [tex]\(x\)[/tex] and [tex]\(a \cdot b = c\)[/tex] (the constant term).
- Given \(a + b = 3\), let's find possible values for a and b that would yield a correct factorization:
- Consider [tex]\(a = 1\), \(b = 2\):[/tex] Then [tex]\(a \cdot b = 1 \times 2 = 2\),[/tex] which is different from the given constant \(c\).
- Consider [tex]\(a = -2\), \(b = -5\):[/tex] Then [tex]\(a \cdot b = -2 \times -5 = 10\),[/tex] suggesting that [tex]\(c = 10.[/tex]
- Consider [tex]\(a = 5\), \(b = 2\):[/tex] Then [tex]\(a \cdot b = 5 \times 2 = 10\),[/tex] also suggesting [tex]\(c = 10\).[/tex]
Considering this process, the correct answer that would allow for complete factorization of the given expression [tex]\(x^2 - 3x + c\)[/tex] is 10:
Thus, the factorization of [tex]\(x^2 - 3x + 10\)[/tex] results in [tex]\((x - 2)(x - 5)\),[/tex] suggesting that the constant term in question should be 10.
The complete question is : Which constant term in the expression [tex]\(x^2 - 3x + c\)[/tex] would allow it to be completely factored? Consider the possible values of c and determine which would result in a fully factored form. Options: -10, 0, 10.
The correct constant term that completes the factoring of the expression [tex]\( x^2 - 3x + \ ? \)[/tex] is 10.
To find the constant term that completes the factoring of the quadratic expression [tex]\( x^2 - 3x + \ ? \)[/tex], we can follow these steps:
Understand that when factoring a quadratic expression of the form [tex]\( ax^2 + bx + c \)[/tex], we are looking for two numbers that multiply to ( ac ) and add to ( b ).
In our case, ( a = 1 ), ( b = -3 ), and ( c ) is the constant term we're looking for.
Since ( a ) is 1, we need to find two numbers that multiply to ( c ) and add up to ( -3 ).
These two numbers are the factors of ( c ).
Given that the constant term ( c ) is the term that doesn't include ( x ), it will be the product of these two factors.
To find the constant term, we can factorize the expression ( ac ), where ( a = 1 ) and ( c ) is the constant term.
Once we find the factors of ( c ), we can test different values until we find the correct one that makes the expression factorable.
Let's start by factoring ( ac ):
Since ( a = 1 ), and ( b = -3 ), we have ( ac = c ).
Given that the product of the factors should be ( c ), and the factors should add up to ( -3 ), we can find the factors of ( c ) by trial and error.
Let's try different values of ( c ) and see which one works:
If ( c = 10 ), the factors of ( c ) would be 1 and 10. However, 1 + 10 = 11, not -3.
If ( c = -10 ), the factors of ( c ) would be -1 and 10. However, -1 + 10 = 9, not -3.
If ( c = -10 ), the factors of ( c ) would be 1 and -10. However, 1 + (-10) = -9, not -3.
If ( c = 10 ), the factors of ( c ) would be -1 and -10. However, -1 + (-10) = -11, not -3.
If ( c = 0 ), the factors of ( c ) would be 0 and 0. However, 0 + 0 = 0, not -3.
Based on these trials, we see that none of the values satisfy the condition of adding up to -3.
Therefore, the correct constant term that completes the factoring of the expression [tex]\( x^2 - 3x + \ ? \)[/tex] is 10.
Q10 Q6.) Find a set of parametric equations for the line that passes through the given points
Use synthetic division and the Remainder Theorem to find P(a)
P(x)=x^4+3x^3-6x^2-10x+8 ; a=2
28
–16
2
4
Using synthetic division and the Remainder Theorem, we find that P(a) for P(x) when a=2 is P(2)=4, which is the remainder of the synthetic division.
To find P(a) for the given polynomial P(x)=x⁴+3x³-6x²-10x+8 when a=2, we can use synthetic division. The Remainder Theorem states that the remainder of the division of a polynomial by a linear divisor (x - a) is equal to P(a).
Let's perform the synthetic division:
Write down the coefficients of P(x): 1, 3, -6, -10, 8.Write the value of a below the synthetic division bar: 2.Bring down the leading coefficient: 1.Multiply this coefficient by a and write the result below the next coefficient: 1 * 2 = 2. Add this to the next coefficient: 3 + 2 = 5.Repeat this process for all coefficients.The synthetic division should look like this:
2 | 1 3 -6 -10 8The final number in the bottom row is the remainder, which is also P(2). So, P(2)=4.
A marketing firm conducts a survey to find out how many people use a product. One hundred people are contacted for the study and fifteen confirm that they use the product. Which describes the survey?
The marketing firm is going forward in time and observing groups sharing common factors.
The marketing firm is applying some treatment to the subjects and then proceeding to observe its effect on the subjects.
The marketing firm is observing and measuring specific characteristics of the subjects, but is not attempting to modify the subjects.
The marketing firm is going back in time to collect data over some past period
Final answer:
The marketing firm's survey, where a sample of people is asked if they use a product, is an example of survey research aimed at observing and measuring characteristics without attempting modification. This cross-sectional study helps understand consumer behavior. (Option C)
Explanation:
The survey described by the marketing firm where one hundred people are contacted and fifteen confirm that they use the product is an example of survey research, which is a quantitative research method. In this survey, the firm is observing and measuring specific characteristics of the subjects, but not attempting to modify the subjects. This type of survey can help the firm in understanding consumers' preferences, tastes, attitudes, and behaviors regarding the use of the product. They are not applying any treatment to the subjects or observing them over time as would be done in a longitudinal study. Instead, they're conducting a cross-sectional survey at a single point in time. (Option C)
You own a coffee shop where a cup of coffee costs $2.10. Your cost on the cup of coffee is $0.30. Calculate the margin per cup of coffee.
A quadratic equation is shown below: 3x^2 − 15x + 20 = 0 Part A: Describe the solution(s) to the equation by just determining the radicand. Show your work. Part B: Solve 3x^2 + 5x − 8 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used.
The quadratic equation 3x² − 15x + 20 = 0 has a radicand of -15, indicating no real solutions. The equation 3x² + 5x − 8 = 0 can be solved using the quadratic formula, yielding solutions of x = 1 and x = -8/3.
For the quadratic equation 3x² − 15x + 20 = 0, the radicand can be found as part of the quadratic formula process, which is b^2 - 4ac. Here, a=3, b= -15, and c=20. Substituting these values in, we get the radicand as (-15)² - 4(3)(20) = 225 - 240 = -15. Since the radicand is negative, this indicates that the equation has no real solutions; the solutions are complex numbers.
To solve the equation 3x² + 5x − 8 = 0, we will use the quadratic formula, x = −b ± √(b^2 - 4ac) / (2a), since we have a quadratic with a, b, and c all non-zero. Substituting a=3, b=5, and c= -8, we find the radicand to be (5)² - 4(3)(-8) = 25 + 96 = 121. Calculating further, x = (-5 ± √121) / 6, which simplifies to x = (-5 ± 11) / 6. Thus, we have two solutions: x = (11 - 5) / 6 = 1 and x = (-5 - 11) / 6 = -16/6 = -8/3.
For a test of upper h 0h0: pequals=0.50, the sample proportion is 0.470.47 based on a sample size of 100. use this information to complete parts (a) through (c) below.
(a) Calculate the standard error of the sample proportion.
(b) Determine the z-score for the sample proportion.
(c) Find the p-value for the given z-score.
To find the standard error, z-score, and P-value for the given test, follow these steps:
(a) Calculate the standard error using the formula:
[tex]\[ \text{Standard error} = \sqrt{\frac{p \times (1 - p)}{n}} \][/tex]
Given [tex]\( p = 0.50 \) (the population proportion) and \( n = 100 \) (the sample size), substitute these values into the formula:[/tex]
[tex]\[ \text{Standard error} = \sqrt{\frac{0.50 \times (1 - 0.50)}{100}} \]\[ = \sqrt{\frac{0.50 \times 0.50}{100}} \]\[ = \sqrt{\frac{0.25}{100}} \]\[ = \sqrt{0.0025} \]\[ = 0.05 \][/tex]
(b) Calculate the z-score using the formula:
[tex]\[ \text{z-score} = \frac{\text{sample proportion} - \text{population proportion}}{\text{standard error}} \]Given that the sample proportion is \( 0.47 \), substitute this value along with the previously calculated standard error into the formula:\[ \text{z-score} = \frac{0.47 - 0.50}{0.05} \]\[ = \frac{-0.03}{0.05} \]\[ = -0.6 \][/tex]
(c) Finally, find the P-value using a z-table or statistical software corresponding to the calculated z-score. The P-value represents the probability of observing a sample proportion at least as extreme as the one obtained, assuming the null hypothesis is true.
ΔABC and ΔXYZ are similar triangles. If BC = x + 7, AC = x + 6, YZ = 4 − x, and XZ = 3 − x, find the value of x.
Answer:
-3/2 or -1.5
Step-by-step explanation:
-3/2 = -1.5
BC = -1.5 + 7 = 5.5
AC = -1.5 + 6 = 4.5
YZ = 4 - (-1.5) = 5.5
XZ = 3 - (-1.5) = 4.5
what is the equation for horizontal line through the point (0,3)?
The probabilities for four different simple events are given. Of the four, which is MOST LIKELY to occur?
A. 0.5
B. 0.9
C. 0.35
D. 0.0100
Determine the number of possible solutions for a triangle with A= 30 a=20 and b=16
There is a unique solution for a triangle with A=30 degrees, a=20 units, and b=16 units. By using the Law of Sines, sin(B) is computed as 0.4. Since there is no obtuse angle with the same sine, there is only one possible angle B, leading to one possible triangle.
To determine the number of possible solutions for a triangle with A=30 degrees, a=20 units (side opposite angle A), and b=16 units (another side), we need to apply the Law of Sines and explore the possible cases. According to the Law of Sines:
a/sin(A) = b/sin(B)
For the given values, we have:
20/sin(30 degrees) = 16/sin(B)
Calculating sin(B) gives us:
sin(B) = 16 * sin(30 degrees) / 20 = 8/20 = 0.4
Now, if sin(B) is less than 1, which it is in our case, there are two possible scenarios:
B is acute: There will be one solution for B, meaning B could be angle whose sine is 0.4.
B is obtuse: We must also check if there is a possible obtuse angle that also has a sine of 0.4. However, since the sine function is positive and less than or equal to 1 for angles between 0 and 180 degrees and it's symmetric with respect to 90 degrees, there can't be an obtuse angle with the same sine value as an acute angle.
Therefore, we only have one possible angle B, which implies we have a unique solution for the triangle.
Additionally, we should check whether side b is larger than the altitude from A; otherwise, there would be no solution for the triangle. To do this, we can use the extended Law of Sines to calculate the diameter (D) of the triangle's circumcircle:
D = a / sin(A)
And thus, the altitude (h) from A would be:
h = D * sin(B)
If b > h, we have a valid triangle and a unique solution.
There is exactly one possible solution for a triangle with the given side lengths a = 20, b = 16, and angle A = 30°.
To determine the number of possible solutions for a triangle given the side lengths a, b, and the angle A, we can use the Law of Sines. The Law of Sines states:
[tex]\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\][/tex]
where:
a, b, c - side lengths of the triangle
A, B, C - angles opposite to the respective sides
Given:
A = 30°
a = 20
b = 16
We can find the angle B using the Law of Sines:[tex]\[\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\][/tex]
[tex]\[\frac{20}{\sin(30^\circ)} = \frac{16}{\sin(B)}\][/tex]
sin (B) = [tex]\frac{16 \times \sin(30^\circ)}{20}[/tex]
= [tex]\frac{16 \times 0.5}{20}[/tex]sin
= [tex]\frac{8}{20}[/tex]
= 0.4
To find the angle B, we take the inverse sine:
B = [tex]\sin^{-1}(0.4)[/tex]
B = 23.58°
Now, we can find angle C since the sum of angles in a triangle is 180°:C = 180° - A - B
C = 180° - 30° - 23.58°
C = 126.42°
Now, let's check if the side lengths satisfy the triangle inequality theorem:
a + b > c
20 + 16 >
36 > c
Since 36 is greater than c, the triangle inequality theorem is satisfied.
So, there is exactly one possible solution for a triangle with the given side lengths a = 20, b = 16, and angle A = 30°.
Calculate the taxes based on the following taxable income amounts
I really need help on this fast please!
I need the estimated taxes for each problem at the bottom is the table needed.
1) income of 90,714.00
2) income of 63,036.00
3) income of 220,596.00
4) income of 166,904.00
5) income of 220,770.00
6) income of 97,266.00
7) income of 207,264.00
8) income of 479,956.00
9) income of 511,195.00
10) income of 511,104.00
11) income of 212,817.00
12) income of 435,045.00
13) income of 655,952.00
14) income of 739,334.00
15) income of 322,335.00
Income range Taxes
From To Tax rate. Plus
0.00 8,925.00 10.0%.
8925.00 36,250.00 15.0% 892.50
36,250.00 87,850.00 25.0% 5,437.50
87,850.00 183,250.00 28.0% 21,962.50
183,250.00 398,350.00 33.0% 51,310.00
398,350.00 400,000.00 35.0%. 131,455.50
400,000.000 - 39.6%. 140.000.00
A drink contains 20% cranberry juice and the rest is apple juice. What is the ratio of cranberry juice to apple juice? A.1:20 B.1:4 C.4:1 D.20:1
the circumference of an Oreo is 6.28 inches what is the area of the top cookie?
The circumference of an Oreo is 6.28 inches what is the area of the top cookie?
Solution:
Circumference= 6.28 inches
The formula of circle =2πr
So, 2πr=6.28
2*3.14*r=6.28
6.28r=6.28
To find the value of r, Let us divide by 6.28 on both sides
6.28 r /6.28 =6.28/6.28
r=1
Area of circle=πr²
=π1²
=π
=3.14 inches²
Area of the top cookie=3.14 inches²
What is the change in temperature between -8°c and 3°c ?
A package of three compact discs cost $24.96. How much does each disc cost?
3^2·3^5·3^-3 Please help.
PLEASE PLEASE HELP ASAP
I had 42 ounces of rice. On Monday I bought 58 more ounces of rice. Then I divided all the rice into 10 equal-sized portions for dinner with friends. How many ounces of rice were in each portion? ounces
The student initially had 42 ounces of rice, bought 58 more for a total of 100 ounces. These were divided into 10 portions, so each portion will contain 10 ounces of rice.
Explanation:This problem is a basic arithmetic question. The student starts with 42 ounces of rice and then adds 58 more ounces, for a total of 100 ounces. She then divides these 100 ounces into 10 equal portions. The key here is to understand the concept of division, which basically means splitting up a total amount (100 ounces) evenly into a certain number of parts (10 portions). To do this, simply use the operation of division: 100 divided by 10 equals 10. Thus, each portion will contain 10 ounces of rice.
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What are the zeros of the quadratic function f(x) = 2x2 + 16x – 9?
Answer:
The zeros to the quadratic equation are:
[tex]x= -4+\sqrt{\frac{41}{2}}\\\\x= -4-\sqrt{\frac{41}{2}}[/tex]
Step-by-step explanation:
A quadratic function is one of the form [tex]f(x) = ax^2 + bx + c[/tex], where a, b, and c are numbers with a not equal to zero.
The zeros of a quadratic function are the two values of x when [tex]f(x) = 0[/tex] or [tex]ax^2 + bx +c = 0[/tex].
To find the zeros of the quadratic function [tex]f(x)= 2x^2 + 16x -9[/tex] , we set [tex]f(x) = 0[/tex], and solve the equation.
[tex]2x^2+16x\:-9=0[/tex]
[tex]\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\mathrm{For\:}\quad a=2,\:b=16,\:c=-9:\quad x_{1,\:2}=\frac{-16\pm \sqrt{16^2-4\cdot \:2\left(-9\right)}}{2\cdot \:2}\\\\x=\frac{-16+\sqrt{16^2-4\cdot \:2\left(-9\right)}}{2\cdot \:2}= -4+\sqrt{\frac{41}{2}}\\\\x=\frac{-16-\sqrt{16^2-4\cdot \:2\left(-9\right)}}{2\cdot \:2}= -4-\sqrt{\frac{41}{2}}[/tex]
what is The desired outcomes of a specified event.
Answer:
Favorable Outcomes
Step-by-step explanation:
Graph the parabola 3x^2+6x-24
help me pls i need to pass
The surface area (S) of a rectangular prism is given in terms of its length (L), width (W), and height (H) as
... S = 2(LW +WH + LH)
Substitute your numbers and do the arithmetic.
... S = 2(7.2·10 + 10·6.3 + 7.2·6.3)
... = 2(72 + 63 + 45.36)
... = 2·180.36
... = 360.72 . . . . square inches
Paisley needs 32 in of ribbon for a hair bow. The ribbon is sold by centimeters. Givin that 1 in = 2.54 centimeters, how many centimeters of ribbon does she need.
Work Pls
Thx
If a bank has total deposits of $7,700 and reserves of $2,100: instructions: round to the nearest whole number. (a) what is its current reserve ratio? % (b) how large is its loan portfolio (outstanding loans)? %
Vicente is watching a movie that last 1 hour and 37 minutes you watch 52 minutes of our how many minutes are left in the movie
Can somebody solve this system equation using addition please?
-4x-5y=7
3x+5y=-14
I can’t solve it and it’s due tomorrow for my math class also please show work on how you’ve solved it
-4x - 5y = 7
3x + 5y = -14
You can add these two equations together straightaway since the y-terms have opposite coefficients.
-4x - 5y = 7
3x + 5y = -14
+___________
-x - 0 = -7
-x = -7
x = 7
Substitute 7 for x into either of the original equations and solve algebraically to find y.
3x + 5y = -14
3(7) + 5y = -14
21 + 5y = -14
21 = -14 - 5y
35 = -5y
-7 = y
Finally, check work by substituting both x- and y-values into both original equations.
-4x - 5y = 7
-4(7) - 5(-7) = 7
-28 + 35 = 7
7 = 7
3x + 5y = -14
3(7) + 5(-7) = -14
21 - 35 = -14
-14 = -14
Answer:
x = 7 and y = -7; (7, -7).
Experts/ace/geniuses helppp asapp
Last year 40 people adopted a manatee through Fia's Foundation.This year 30% more people adopted a manatee. How many more people adopted a manatee this year ?
The equilateral triangle shown is rotated about line a. Each side of the triangle measures 20 mm. What shape is created by the rotation and what is the approximate circumference of the base? Circumference of a circle: C = 2πr a cylinder with a circumference of about 63 mm a cylinder with a circumference of about 126 mm a cone with a base circumference of about 63 mm a cone with a base circumference of about 126 mm
The shape and the circumference of the base created by the rotation is; Cone and Circumference of 63 mm
What shape is created by rotation?We are given;
Each side of equilateral triangle = 20 mm
Now, when the triangle is rotated about the line ‘a’ which passes through the midpoint of any of the 3 sides of the equilateral triangle we will get a conical shape.
The distance from the line a which cuts one side of a triangle to one vertex of the triangle is the radius = 10 mm
We know that;
Circumference of a circle = 2πr
Thus;
C = 2 × 3.14 × 10
C = 6.28 × 10
C = 62.8 mm
C = 63 mm
Thus, the approximate circumference of the base is 63 mm.
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