Answer:
k = 4
Step-by-step explanation:
Polynomials:
p(x) = 2x³ - 3x² - kx - 17
x - 3 = 0
x = 3
It is given that when p(x) is divided by (x - 3), it leaves the remainder -2
⇒ p(3) = -2
2(3)³ - 3(3)² - k *3 - 17 = -2
2*27 -3*9 - 3k -17 = -2
54 - 27 - 3k -17 = -2
-3k + 10 = -2
-3k = -2 - 10
-3k = -12
k = (-12) ÷ (-3)
[tex]\sf \boxed{k = 4}[/tex]
Use the quadratic formula to solve: 3x² + 4x + 2 = 0.
What is the value of the discriminant?
What does this tell you about the solutions to the equation?
Answer:
[tex]x=-\frac{2}{3}+\frac{\sqrt{2}}{3}i\\\\x=-\frac{2}{3}-\frac{\sqrt{2}}{3}i[/tex]
Step-by-step explanation:
The Quadratic Formula gives us the roots of a quadratic and is defined as: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
The discriminant is the part in the square root which is: [tex]b^2-4ac[/tex]
if the discriminant of the quadratic is zero, then the quadratic only has one distinct zero with a multiplicity of 2.
To see why, let's plug in zero for the discriminant: [tex]x=\frac{-b\pm\sqrt{0}}{2a}[/tex]
The square root of zero is just going to be zero: [tex]x=\frac{-b\pm0}{2a}[/tex]
Adding or subtracting zero doesn't change the value: [tex]x=\frac{-b}{2a}[/tex]
As you can see here it doesn't matter if we take the positive or negative solution, we'll get one value.
If the discriminant of the quadratic is positive, then the quadratic will have two distinct real solutions.
Well the solutions are real because the square root has real solutions for positive numbers. The reason we have two distinct solutions is because the square root of a positive number will have a positive and negative solution. These square roots will have a non-zero value and will affect the value of the numerator based on whether you take the negative or positive solution giving you two distinct solutions.
If the discriminant of the quadratic is negative, then the quadratic will have two distinct complex solutions.
To clarify, real numbers are technically complex numbers, since complex numbers are just: [tex]a+bi[/tex], where a=real part, and b=imaginary part. We could use this to represent any real number where b=0.
So when I'm saying "complex solution" I mean a complex solution where b does not equal zero, so there is a non-zero imaginary component to the complex solution.
The reason we have a complex solution in the first place is because we have a negative number in the square root. The square root only has real solutions for positive numbers, since any real number squared will give you a positive number regardless of the sign (positive or negative). The only number that is negative when squared is an imaginary number.
The reason we have two distinct solutions is actually the same reason we have two distinct solutions when the discriminant is positive. Since the square root will yield a non-zero number, so when we add it or subtract it to b, we will get to different values for the numerator, giving us two different values, or two different solutions.
So now let's solve the equation: [tex]3x^2+4x+2=0[/tex]
We have it in standard form where: [tex]ax^2+bx+c=0[/tex]
we want to use the Quadratic Formula: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
Plugging in values we get:
[tex]a=3,\ b=4,\ c=2\\\\x=\frac{-4\pm\sqrt{4^2-4(3)(2)}}{2(3)}\\\\x=\frac{-4\pm\sqrt{16-24}}{6}\\\\x=\frac{-4\pm\sqrt{-8}}{6}[/tex]
From here you'll notice the discriminant is -8, and thus we complex solutions since it's negative. We can first break up this square root into multiple square roots using the fact that: [tex]\sqrt{ab}=\sqrt{a}*\sqrt{b}[/tex]
and since -8 can be defined as: -1 * 8, we can split it up like this
[tex]x=\frac{-4\pm\sqrt{-1}*\sqrt{8}}{6}\\\\x=\frac{-4\pm i\sqrt{8}}{6}[/tex]
We can actually simplify the square root a bit more, since we can once more use the identity: [tex]\sqrt{ab}=\sqrt{a}*\sqrt{b}[/tex], to simplify the radical. The number 8 can be represented as 4 * 2, and sqrt(4) can be simplified easily.
[tex]x=\frac{-4\pm i *\sqrt{4}*\sqrt{2}}{6}\\\\x=\frac{-4\pm 2i\sqrt{2}}{6}\\\\x=\frac{(-2 * 2)\pm2i\sqrt{2}}{(3*2)}\\\\x=\frac{-2\pm i\sqrt{2}}{3}[/tex]
Now let's take the positive and negative solution to get the two distinct complex solutions. You can start with either but I'll just do the positive first.
Positive Solution:
[tex]x=\frac{-2+ i\sqrt{2}}{3}[/tex]
You could leave it like this, but generally whenever we have a complex number we want it in the form: [tex]a+bi[/tex] where the real and imaginary part are separate, so we can just distribute the division of 3
Positive Solution Continued:
[tex]x=\frac{-2+ i\sqrt{2}}{3}\\\\x=\frac{-2}{3}+\frac{\sqrt{2}}{3}i[/tex]
I also moved the "i" outside the fraction, so it's a bit easier to see the value of "b"
Negative Solution:
[tex]x=\frac{-2-i\sqrt{2}}{3}\\\\x=\frac{-2}{3}-\frac{\sqrt{2}}{3}i[/tex]
Please help will give brainliest B
Yuto left his house at 10 a.m. to go for a bike ride. By the time Yuto’s sister Riko left their house, Yuto was already 5.25 miles along the path they both took. If Yuto’s average speed was 0.25 miles per minute and Riko’s average speed was 0.35 miles per minute, over what time period in minutes, t, starting from when Riko left the house, will Riko be behind her brother?
The time period that Riko would be behind is going to be the inequality 0 ≤ t < 52.5 minutes .
How to solve the inequalityHow to solve for the period t that Riko would be behind her brother
The question says that at the time he left the house, Yuto was 5.25 miles ahead of her.
The amount of miles that she would have gained on Yuto within this period would be 0.35 - 0.25 = 0.1 miles on Yuto
Therefore at every mile she would be closing in on 0.1 miles
For her to close the gap that is made here, it would be 5.25 / 0.1
= 52.5 minutes
Therefore we would say that The time that Riko would be behind is going to be 0 ≤ t < 52.5 minutes .
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Ella buys a hat in Sweden for 50 SEK( Swedish Krona).
Find the VAT paid by Ella.
Answer:
12.5 SEK
Step-by-step explanation:
VAT percentage in Sweden is 25% so, 25% of 50 = 12.5
Find the missing side length, x. (image)
The missing side of the triangle is 7.2 units
How to find the missing side of a triangle?ABC is a triangle.
AD bisects(cut into half) the angle ∠BAC.
Therefore,
AB / BD = AC / DC
Hence,
AB = 12
BD = x
AC = 20
DC = 12
Therefore,
12 / x = 20 / 12
cross multiply
12 × 12 = 20x
144 = 20x
divide both sides by 20
x = 144 / 20
Therefore,
x = 7.2 units.
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280 -8 (-14+x)
What’s the solution??
[tex]=280-8(-14)-8(x)\\=280+112-8x\\=392-8x[/tex]
Attached is the solution. be alert -8 multiplies everything in the brackets
Choose the improper fraction that is equivalent to the mixed numbers 4 3/4
Answer:
d) 19/4
Step-by-step explanation:
Given mixed number,
→ 4 3/4
Now the improper fraction is,
→ 4 3/4
→ 19/4
Hence, option (d) is correct.
each year, students in an elementary school take a standardizzed math test at the end of the school year. for a class of fourth-graders, the average score was 55.1 with a standard deviation of 12.3. in the third grade, these same students had an average score of 61.7 with a standard deviation of 14.0. the correlation between the two sets of scores is r
Answer:
y = 0.835x + 3.58
Step-by-step explanation:
A shape is shown on the graph: A coordinate grid is shown from positive 8 to negative 8 on the x-axis and from positive 8 to negative 8 on the y-axis. A triangle is shown on ordered pair 2, negative 2 and 4, negative 2 and 2, negative 6. Which of the following is a reflection of the shape? (5 points) a A coordinate grid is shown from positive 8 to negative 8 on the x-axis and from positive 8 to negative 8 on the y-axis. A triangle is shown on ordered pair 2, 2 and 4, 2 and 2, 6. b A coordinate grid is shown from positive 8 to negative 8 on the x-axis and from positive 8 to negative 8 on the y-axis. A triangle is shown on ordered pair negative 2, negative 2 and negative 4, negative 2 and negative 4, negative 6. c A coordinate grid is shown from positive 8 to negative 8 on the axis and from positive 8 to negative 8 on the y-axis. A triangle is shown on ordered pair 2, 2 and 2, 4 and 6, 2. d A coordinate grid is shown from positive 8 to negative 8 on the axis and from positive 8 to negative 8 on the y-axis. A triangle is shown on ordered pair negative 2, negative 4 and negative 6, negative 4 and negative 6, negative 2.
The option which gives the reflection of the triangle (shape) with vertices, (2, -2), (4, -2), and (2, -6) is option a
a. A triangle is shown on ordered pair (2, 2), (4, 2), (2, 6)
What is a reflection transformation?A reflection transformation is one in which the line across which the object is reflected acts like a mirror, such that the points are swapped across the line of reflection.
The coordinates of the vertices of the triangle on the coordinate grid are;
(2, -2), (4, -2), and (2, -6)Required; The coordinates of a reflection of the shape (the triangle)
Solution; The coordinates of a point (x, y) following a reflection across the y–axis is (-x, y)
The vertices of the image of the triangle following a reflection across the y–axis are therefore;
(2, -2) → (-2, -2)(4, -2) → (-4, -2)(2, -6) → (-2, -6)A reflection of the point (x, y) across the x–axis gives the point (x, -y)
A reflection of the triangle across the x–axis therefore gives;
(2, -2) → (2, 2)(4, -2) → (4, 2)(2, -6) → (2, 6)A reflection of the point (x, y) across the line y = x, gives (y, x)
The vertices of the triangle following a reflection across the line y = x are therefore;
(2, -2) → (-2, 2)(4, -2) → (-2, 4)(2, -6) → (-6, 2)A reflection of the point (x, y) across the line y = -x, gives (-y, -x)
The vertices of the triangle following a reflection across the line y = -x are therefore;
(2, -2) → (2, -2)(4, -2) → (2, -4)(2, -6) → (6, -2)The coordinates of the triangle in the options are;
a) (2, 2), (4, 2), (2, 6)
b) (-2, -2), (-4, -2), (-4, -6)
c) (2, 2), (2, 4), (6, 2)
d) (-2, -4), (-6, -4), (-6, -2)
The correct option is therefore option a with points on the image of the triangle of (2, 2), (4, 2), (2, 6) which corresponds with the coordinates of the triangle following a reflection across the x–axis
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AP Stats work, really confused please help asap!! Thank you
Answer:
The points scored for each game are approximately normally distributed. The numbers distributed are in order of value from the least to greatest, making it easy to get the medium or mean.
Step-by-step explanation:
Two car owners are in need of car repairs. Hazel will need to pay the mechanic $1 per minute for labor, plus $40 to cover the cost of new parts. Megan will need to pay $10 for parts and $4 per minute for labor. Depending on how long each repair takes, the two jobs might end up costing the same amount. How much time would that take?
Write a system of equations, graph them, and type the solution.
Answer: 10 minutes
Step-by-step explanation:
Solve for the variable k
The equation m/k = x solved for k will be k = m / x.
What are variables?variable, in algebra, a symbol (usually a letter) that represents an unknown numerical value in an equation. Commonly used variables are x and y (real unknowns), z (complex unknowns), t (time), r (radius), and s (arc length).
There are two types of variables in math:
dependent independent variablesIn mathematics, a variable is an alphabet or term that represents an unknown number or an unknown value or an unknown quantity. Variables are used especially in algebraic expression or algebra. For example, x 9=. is a linear equation where x is a variable, where 9 and are constants.
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asap !! 50 points!!!!!!!!
Which is bigger? Show your work WITHOUT EVALUATING.
A = 3^2/2^3
or
B = 9^4/4^6
please help, it doesn’t let me give 100 points :/
Answer:
B is bigger
Step-by-step explanation:
a) 3^2 / 2^3
3*3 / 2*2*2
9/8
b) 9^4 / 4^6
9*9*9*9/4*4*4*4*4
6561 / 4096
1.601807
screenshot underneath
Simplify them
Answer =11
a= bc-3c-3b/6c
look at the screenshots for steps
It is -24° tonight, but the weather reporter predicted it would be 16° warmer tomorrow.
What will the temperature be tomorrow?
Answer:
-8°
Step-by-step explanation:
-24+16=-8
Does (-1,3) make the equation y=-3x. true?
Answer: Yes, the point (-1, 3) makes the equation true.
Step-by-step explanation:
You would need to plug in the coordinates to the following places in the equation.
y=-3x
(3) = -3(-1)
3 = 3
Hope this helps!
Have a great day! :)
i iam in 4th grade, i need to know 5×395=5×(400×-5)
11975 is value of multiplication .
What does multiplication mean in math?
Multiplication in mathematics is a technique for determining the sum of two or more numbers. It is a fundamental operation in mathematics that we perform frequently.When a single number is added up several times, it is said to be multiplied. A 5 multiplied by a 4 results in 20 (5 + 5 + 5 + 5). The number five was taken, and four additions were made. Due to this, multiplication is occasionally referred to as "times."5×395=5×(400×-5)
1975 = 5 * ( - 2000 )
1975 = - 10000
= 10000 + 1975
= 11975
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what is the solution for this inequality -13x<65
Step-by-step explanation:
-13x < 65
-13x * -1 < 65 * -1
13x > -65
13x/13 > -65/13
x > -5
Hope this helps! :)
(X+3)/6=5/4 find x
A-27/4
B-18
C-9/8
D-9/2
Answer:
d
Step-by-step explanation:
(x+3)/6 = 5/4
butterfly method
4(x+3) = 5×6
4x+12 = 30
4x = 30-12
4x = 18
x = 18/4
x = 9/2
please help i need this in an hour is it right or not
Answer:
c 6√5 - 4√5
Step-by-step explanation:
your answer is correct
0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 find the interquartile range
The interquartile range for the given set of numbers {0,2,4,6,8,10,12,14,16,18,20} is 12 .
The interquartile range(IQR) can be calculated by the formula
IQR = Q₃ - Q₁
where Q₃ is the third quartile
and Q₁ is the first quartile
In the question,
the set of numbers is ascending order is given as
0,2,4,6,8,10,12,14,16,18,20
Q₃ means median of the upper half
the upper half of the data is 12,14,16,18,20
the median of the upper half is 16 ,
So Q₃ = 16 .
Q₁ means median of the lower half
the lower half of the data is 0,2,4,6,8
the median of the lower half is 4 ,
So Q₁ = 4 .
Hence , the IQR = Q₃ - Q₁ = 16-4 = 12 .
So, IQR for given set of numbers is 12.
Therefore , the interquartile range for {0,2,4,6,8,10,12,14,16,18,20} is 12 .
The given question is incomplete , the complete question is
Find the Interquartile range for the given set of numbers 0,2,4,6,8,10,12,14,16,18,20 .
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Pls help step by step
Answer:
25
Step-by-step explanation:
the tens cancle each other out
The following table shows the number of U.S. coins produced by the Denver Mint from the years
2009 to 2011.
Denver Mint Circulating Coin Production
(in millions)
Year Pennies
Nickels Dimes Quarters
If a coin produced by the Denver Mint in 2010 is selected at random, what is the probability that
the coin is a dime?
Answer: 0.187
Step-by-step explanation:
I just did the question
I only need the answer for 18 please
The equation of the line parallel to 2x + 3y = 18 and passing through point ( 6, 4) is 3y + 2x = 24.
Consider the line 2x + 3y = 18,
The equation of the line in slope-intercept form can be written in the form y = mx + b, where m is the slope of the line.
Therefore,
2x + 3y = 18
Subtracting 2x from each side
2x + 3y - 2x = 18 - 2x
3y = 18 - 2x
Dividing each side by 3.
y = 6 - ( 2/3 )x
y = - (2/3)x + 6
Hence, the slope of the line is - 2/3.
Now, if two lines are parallel then they have the same slope.
Now, the line parallel to 2x + 3y = 18 passes through ( 6,4).
Therefore, using the slope-intercept form:
y = mx + b
4 = ( -2/3 ) × 6 + b
4 = - 4 + b
b = 8
Therefore, the equation of the line parallel to 2x + 3y = 18 is:
y = ( - 2/3 )x + 8
3y = - 2x + 24
3y + 2x = 24
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6(a) +4 a=1/3
PLEase HELP ME
ASAPP!!!
Answer:
a=1/30
Step-by-step explanation:
6a+4a=1/3
10a=1/3
30a=1
a=1/30
1/30
is the anwser i think.i might be wront tho
Find the slope and write an equation of the line which passes through the points.
b
(−3, 9) and (−3, −5)
Answer:
Slope: 0
Equation : y = 9
Plsss mark as Brainliest
Slope:
-5-9/-3+3
=-14/0
= 0
Equation:
input gradient and pair of x and y valuesin y = mx +c
9= 0(-3)+c
9 = 0 +c
c = 9
input gradient and c:
y = 9
Solve the proportion.
I WILL USE CROSS MULTIPLICATION
[tex]63(4x) = 84(12) \\ 63(4x) = 1008 \\ \frac{63(4x)}{63} = \frac{1008}{63} \\ 4x = 16 \\ \frac{4x}{4} = \frac{16}{4} \\ x = 4[/tex]
ATTACHED IS THE SOLUTION
If −3+ n≤ 25, which inequality represents the possible range of values for 4n-12?
A) 4n-12≤ -100
B) 4n-12≤ 100
C) 4n-122-100
D) 4n-12≥100
The range is 4n - 12≤100.
The inequality is -3 + n≤25. From this, we can find the range of n.
n≤25 + 3
n≤28
Now we have to find the range of 4n - 12. For this, we have to ≤multiply both sides with 4 and then subtract it from 12, in the range of n.
That is;
4n≤ 28×4
4n - 12 ≤112 -12
4n - 12 ≤100
Therefore the range is 4n - 12≤ 100.
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How many times larger than 6 x 10^-2 is 24 x 10^6?
Is our divisor a whole number? 13.5 divided by 0.03 [Powers of 10 - What should you multiply by to get a whole number?]
No, the divisor 0.03 is not a whole number, we should multiply by 10² to get a divisor a whole number.
As given in the question,
Given expression:
13.5 divided by 0.03
Dividend of the given expression = 13.5
Divisor of the given expression = 0.03
No, the divisor 0.03 is not a whole number.
Power of 10 which should be multiply by 0.03 to get a whole number
0.03 = 3 /100
(3/100) should get multiply by 100
100 = 10²
0.03 ×10² =3 whole number
Therefore, the divisor 0.03 is not a whole number, we should multiply by 10² to get a divisor a whole number.
The complete question is:
Is our divisor a whole number? 13.5 divided by 0.03 . By what powers of 10 should you multiply by to get a whole number?
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