Answer:
2
Step-by-step explanation:
So I'm going to use vieta's formula.
Let u and v the zeros of the given quadratic in ax^2+bx+c form.
By vieta's formula:
1) u+v=-b/a
2) uv=c/a
We are also given not by the formula but by this problem:
3) u+v=uv
If we plug 1) and 2) into 3) we get:
-b/a=c/a
Multiply both sides by a:
-b=c
Here we have:
a=3
b=-(3k-2)
c=-(k-6)
So we are solving
-b=c for k:
3k-2=-(k-6)
Distribute:
3k-2=-k+6
Add k on both sides:
4k-2=6
Add 2 on both side:
4k=8
Divide both sides by 4:
k=2
Let's check:
[tex]3x^2-(3k-2)x-(k-6) \text{ with }k=2[/tex]:
[tex]3x^2-(3\cdot 2-2)x-(2-6)[/tex]
[tex]3x^2-4x+4[/tex]
I'm going to solve [tex]3x^2-4x+4=0[/tex] for x using the quadratic formula:
[tex]\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\frac{4\pm \sqrt{(-4)^2-4(3)(4)}}{2(3)}[/tex]
[tex]\frac{4\pm \sqrt{16-16(3)}}{6}[/tex]
[tex]\frac{4\pm \sqrt{16}\sqrt{1-(3)}}{6}[/tex]
[tex]\frac{4\pm 4\sqrt{-2}}{6}[/tex]
[tex]\frac{2\pm 2\sqrt{-2}}{3}[/tex]
[tex]\frac{2\pm 2i\sqrt{2}}{3}[/tex]
Let's see if uv=u+v holds.
[tex]uv=\frac{2+2i\sqrt{2}}{3} \cdot \frac{2-2i\sqrt{2}}{3}[/tex]
Keep in mind you are multiplying conjugates:
[tex]uv=\frac{1}{9}(4-4i^2(2))[/tex]
[tex]uv=\frac{1}{9}(4+4(2))[/tex]
[tex]uv=\frac{12}{9}=\frac{4}{3}[/tex]
Let's see what u+v is now:
[tex]u+v=\frac{2+2i\sqrt{2}}{3}+\frac{2-2i\sqrt{2}}{3}[/tex]
[tex]u+v=\frac{2}{3}+\frac{2}{3}=\frac{4}{3}[/tex]
We have confirmed uv=u+v for k=2.
Find the range of the following set of data.
13. 11, 4,5,6, 9, 10, 12, 15, 16
Answer:
12
Step-by-step explanation:
First put the set in order from least to greatest:
4, 5, 6, 9, 10, 11, 12, 15, 16
To find the range you subtract the largest number by the smallest number.
4, 5, 6, 9, 10, 11, 12, 15, 16
So 16 - 4 = 12
So the range is 12.
Answer:
12
Step-by-step explanation:
The greatest value in the data is 16.
The lowest value in the data is 4.
The range is the difference between the highest and the lowest values.
range = 16 - 4 = 12
What are the values of a and b?
a = 14, b = 6
a = 14, b = 8
a = 17, b = 6
a = 17, b = 8
According to the question the values of a and b are 17 and 6 respectively
What is a kite?A kite is a quadrilateral that has 2 pairs of equal-length sides and these sides are adjacent to each other.
According to the Definition:
Sides FJ = HJ ( from the diagram)
Hence (3b + 6) = 24cm
3b = 24 - 6
3b = 18
b = 6
Sides FG = GH (from the diagram)
(2a - 4) = 30
2a = 30 + 4
a = 34/2
a = 17
Hence the value of a and b are 17 and 6 respectively.
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Answer: C
Step-by-step explanation:
Which expression is equal to a + (b + c)?
(a + b) + c
(a + b) ⋅ c
a + bc
b + ac
Answer:
The correct option is (a+b)+c
Step-by-step explanation:
The correct option is (a+b)+c
According to the Associative property for Addition:
a+(b+c)=(a+b)+c
The associative property states that you can add or multiply regardless of how the numbers are grouped.
.If you are adding or multiplying it does not matter where you put parenthesis.
Thus the correct option is A....
[tex]\huge{\boxed{(a+b)+c}}[/tex]
For example, [tex]2+(3+4)=2+7=9[/tex].
If we group the terms differently, it doesn't matter. [tex](2+3)+4=5+4=9[/tex]
This is called the associative property of addition. As long as you still have all of the terms, and there are no other operations (subtraction, multiplication, etc.), the final answer will remain the same no matter how the terms are grouped.
5 / 3 + 43 / 9 what is the answer
Answer:
58/9
Step-by-step explanation:
5/3 +43/9
Lets take the L.C.M first
The L.C.M would be 9
Solve the term by taking 9 as L.C.M
=15+43/9
Add the numerator.
=58/9
The answer is 58/9 ....
Find the value of X in the picture.
Answer:
The measure of the arc x is 130°
Step-by-step explanation:
we know that
The semi-inscribed angle is half that of the arc it comprises
so
65°=(1/2)[arc x]
solve for x
arc x=(2)(65°)=130°
What is the complex conjugate?
Answer:
B. -3 - 5i
Step-by-step explanation:
To find the complex conjugate of a complex number, just change the sign of the imaginary part.
The complex conjugate of a + bi is a bi.
In this case, the complex conjugate of 3 + 5i is -3 - 5i.
A complex conjugate of a complex number a + bi is a - bi. It is obtained by changing the sign of the imaginary part. It is useful for performing mathematical operations with complex numbers.
Explanation:The complex conjugate of a complex number is defined as changing the sign of the imaginary part of that number. If the complex number is expressed in the form a + bi, where a and b are real numbers and 'i' is the square root of -1 representing the imaginary unit, its complex conjugate is a - bi. The definition of a complex conjugate is very important in the study of complex numbers because it allows for the multiplication and division of complex numbers in a way that results in real numbers. We use this conjugate to calculate the magnitude or modulus of a complex number as well.
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f(x)=5x-10 and g(x)=x-16
Which of the following represents h(x)=F(x)+g(x)
Answer:
6x - 26
Step-by-step explanation:
f(x) + g(x) = 5x - 10 + x - 16 ← collect like terms
= 6x - 26
Answer:
B
Step-by-step explanation:
The ratio of Holly's age to that her aunt is 4:9. When holly was born her aunt was 15 years old. How old is holly's aunt now?
Answer:
27 years
Step-by-step explanation:
Let Holly's age be h and Aunt's age be a. We can setup 2 equations and solve.
1. [tex]\frac{h}{a}=\frac{4}{9}[/tex]
2. a - 15 = h
We can plug in equation 2 into equation 1 to solve for aunt's age:
[tex]\frac{h}{a}=\frac{4}{9}\\\frac{a-15}{a}=\frac{4}{9}\\9(a-15)=4a\\9a-135=4a\\5a=135\\a=\frac{135}{5}=27[/tex]
Holly's aunt's age is 27
Which of the following is the measure of AXYif ray xy bisects AXB,
which measures 110°
Answer:
The measure of angle AXY is 55°
Step-by-step explanation:
we know that
If ray XY bisects AXB
then
∠AXY=∠YXB=∠AXB/2
therefore
∠AXY=110°/2=55°
see the attached figure to better understand the problem
Use function notation to write a recursive formula to represent the sequence: 3, 6, 9, …
A. f(n) = f(n − 1) + 3
B. f(n) = f(n − 1) + 2
C. f(n) = f(n − 1) ⋅ 3
D. f(n) = f(n − 1) ⋅ 2
Final answer:
The recursive formula for the sequence 3, 6, 9, ... is represented by option A: f(n) = f(n - 1) + 3, which states that each term is 3 more than the previous term, starting with f(1) = 3. Option a
Explanation:
To write a recursive formula for the sequence 3, 6, 9, ..., we need to observe the pattern of the sequence. Since each term increases by 3 from the previous term, we can express this as:
f(n) = f(n − 1) + 3 for n > 1,
with a starting value f(1) = 3. This represents the first term in the sequence. Therefore, the correct answer is A: f(n) = f(n − 1) + 3.
The other options suggest either multiplying the preceding term by 2 or 3, or adding 2 to the preceding term, but those operations do not describe the given sequence. Option a
What is the solution to 3|–3x + 9| = –18?
Answer:
X=5
Step-by-step explanation:
i hope this helps
Answer:
equation has no solutions
Step-by-step explanation:
3|–3x + 9| = –18 (divide both sides by 3)
|–3x + 9| = –6
because by definition, for any value a, |a| must be non-negative
hence |–3x + 9| must give a value that is greater or equal zero
because the right side of the equation is a negative integer, hence the equation has no solutions.
which is a step in the process of calculating successive discounts of 8% and 10% on a $50 item
Answer:
The process of calculating successive discounts of 8% and 10% on a $50 item is take 10% of $46.
Step-by-step explanation:
As given
successive discounts of 8% and 10% on a $50 item .
First find out for 8 % discount
8% is written in the decimal form
= 0.08
8 % of $50 item = 0.08 × 50
= $ 4
Price of item after 8% discount = 50 - 4
= $46
First find out for 10 % discount
10% is written in the decimal form
= 0.1
8 % of $48 item = 0.1× 46
= $4.6
Price of item after 8% discount = 46 - 4.6
= $41.4
Therefore in the successive discounts of 8% and 10% on a $50 item is $41.4 .
The final price after applying the successive discounts is $41.40
To calculate successive discounts of 8% and 10% on a $50 item, follow these steps:
First, convert the first discount of 8% to a decimal by dividing 8 by 100, which gives 0.08.
Multiply the original price of the item, $50.00, by 0.08 to find the amount of the first discount: $50.00 x 0.08 = $4.00.
Subtract the first discount from the original price to find the new price: $50.00 - $4.00 = $46.00.
Next, convert the second discount of 10% to a decimal by dividing 10 by 100, which gives 0.10.
Multiply the new price of the item, $46.00, by 0.10 to find the amount of the second discount: $46.00 x 0.10 = $4.60.
Finally, subtract the second discount from the new price to find the final price: $46.00 - $4.60 = $41.40.
The final price after applying the successive discounts is $41.40.
If f(x)=x-3/x and g(x)=5x-4, what is the domain of (f•g)(x)?
Answer:
{x|⅘ ≠ x} → Set-Builder Notation
(-∞, ⅘) ∪ (⅘, ∞) → Interval Notation
Step-by-step explanation:
Plug the g(x) function into the f(x) function for every x you see.
Answer:
domain is the set of all real numbers except 0
Step-by-step explanation:
If f(x)=x-3/x and g(x)=5x-4
(f•g)(x) is f(x) times g(x)
multiply f(x) and g(x)
(f•g)(x) is f(x) times g(x)
Replace f(x) and g(x)
[tex]f(x) \cdot g(x)= (x-\frac{3}{x})(5x-4)[/tex]
Apply FOIL method to multiply it
multiply x inside the parenthesis
[tex]5x^2-4x[/tex]
Multiply the fraction inside the parenthesis
[tex]-5 +\frac{12}{x}[/tex]
[tex]5x^2-4x -5 +\frac{12}{x}[/tex]
We have x in the denominator . Domain is the set of x value for which the function is defined.
When denominator x is 0 then the function is undefined
So domain is the set of all real numbers except 0
What is this anwser? Thanks!
Answer:
v = -90.
Step-by-step explanation:
To isolate v you multiply both sides of the equation by 10:
v *10 / 10 = -9 * 10
v = -90.
Answer:
v = - 90
Step-by-step explanation:
Given
[tex]\frac{v}{10}[/tex] = - 9
Multiply both sides by 10
v = 10 × - 9 = - 90
if f(x)=x/2-3 and g(x)=3x2+x-6, find (f+g)(x)
Answer:
[tex](f+g)(x) = 3x^2+\frac{3x}{2}-9\\or\\(f+g)(x) = \frac{6x^2+3x-18}{2}[/tex]
Step-by-step explanation:
We are given:
[tex]f(x)=\frac{x}{2}-3 \,\, and\,\, g(x) = 3x^2+x-6[/tex]
We need to find [tex](f+g)(x)[/tex]
(f+g)(x) can be found by adding f(x) and g(x)
(f+g)(x) = f(x) + g(x)
[tex](f+g)(x) = \frac{x}{2}-3+(3x^2+x-6) \\(f+g)(x) = \frac{x}{2}-3+3x^2+x-6\\(f+g)(x) = 3x^2+\frac{x}{2}+x-3-6\\(f+g)(x) = 3x^2+\frac{3x}{2}-9\\(f+g)(x) = \frac{6x^2+3x-18}{2}[/tex]
so, (f+g)(x) is:
[tex](f+g)(x) = 3x^2+\frac{3x}{2}-9\\or\\(f+g)(x) = \frac{6x^2+3x-18}{2}[/tex]
Answer:
Step-by-step explanation:
Vince bought 6 boxes of worms to use as bait while fishing with his friends. If each person uses exactly 3/8 of a box of worms, how many people can share the worms.
We are given the following information:
each person uses 3/8 a box
there are 6 boxes
Because each box is [tex]\frac{8}{8}[/tex], and there are 6 boxes, we know that 1 person is [tex]\frac{3}{8*6}[/tex], or [tex]\frac{3}{48}[/tex].
In order to find how many people can use the 6 boxes, we can divide it by 3 (for each use):
48 / 3 = 16
Therefore, 16 people can share the worms.
Hope this helps! :)
Find the perimeter of a triangle with sides that measure 55 cm, 48 cm, and 32 cm..
Answer:
The perimeter of the triangle is 135 cm
Step-by-step explanation:
* Lets explain how to find the perimeter of a triangle
- The perimeter of any triangle is the sum of the lengths of its three sides
- The perimeter of the scalene triangle is P = S1 + S2 + S3
- The perimeter of the isosceles triangle is P = 2S + S1
- The perimeter of the equilateral triangle is P = 3S
* Lets solve the problem
∵ The length of the 1st side is 55 cm
∴ S1 = 55 cm
∵ The length of the 2nd side is 48 cm
∴ S2 = 48 cm
∵ The length of the 3rd side is 32 cm
∴ S3 = 32 cm
∵ The triangle is scalene
∴ P = S1 + S2 + S3
∴ P = 55 + 48 + 32 = 135 cm
* The perimeter of the triangle is 135 cm
Divide 27x3 - 72x2 + 36x by 9x.
Answer:
3x^2 - 8x + 4.
Step-by-step explanation:
Dividing each term by 9x we get:
3x^2 - 8x + 4.
As per cubic equation, the result is [tex](3x^{2}-8x+4)[/tex].
What is a cubic equation?A cubic equation is an equation where the highest power of the variable is 3.
The given linear equation is:
[tex]\frac{27x^{3}- 72x^{2}+ 36x}{9x} \\= \frac{27x^{3}}{9x} -\frac{72x^{2}}{9x}+\frac{36x}{9x}\\ = 3x^{2}-8x+4[/tex]
The result is [tex](3x^{2}-8x+4)[/tex].
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I have been looking at this for hours, please help!
Q.
One liter is approximately equal to 0.26 gallons. Find the volume rounded to the nearest hundredth of a gallon of a container that holds approximately 5.5 liters.
[tex]\bf \begin{array}{ccll} liters&gallons\\ \cline{1-2} 1&0.26\\ 5.5&x \end{array}\implies \cfrac{1}{5.5}=\cfrac{0.26}{x}\implies x=(5.5)(0.26)\implies x=1.43[/tex]
what is 7/8 to a decimal rounded to the nearest eighth
[tex]\dfrac{7}{8}=\dfrac{875}{1000}=0.875[/tex]
Answer:
[tex]\large\boxed{\dfrac{7}{8}=0.875}[/tex]
Step-by-step explanation:
[tex]\bold{METHOD\ 1:}\\\\\dfrac{7}{8}=\dfrac{7\cdot125}{8\cdot125}=\dfrac{875}{1,000}=0.875\\\\\bold{METHOD\ 2:}\\\\\dfrac{7}{8}=7:8\qquad\text{divide 7 by 8 (look at the picture)}\\\\7:8=0.875[/tex]
The weight of 1000 identical samples of a substance are 1 pound. What is the weight of 10 samples?
Answer:
.01 lbs
Step-by-step explanation:
If the weight of 1000 things are 1 pound that means 1 thing has a weight of 1/1000 lbs. So ten things have a weight of 10/1000 lbs or 1/100 lbs or .01 lbs.
A culture started with 1000 bacteria. After 6 hours it few to 1300 bacteria. Predict how many bacteria will be present after 10 hours.
Answer:
1350
Step-by-step explanation:
If a culture started with 1000 bacteria and after 6 hours it few to 1300 bacteria, there would be 1350 present after 10 hours.
1000 to 1100 is an increase of 100 bacteria per 6 hours.
1100+250= 1350
Answer:
1500
Step-by-step explanation:if it grows 300 in 6 hours you can assume that each hour would increase the pop by 50 so 10 hours times 50 it will become 1500.
Find the slope of the line that passes through the points (1, -3) and (2, -1). a.) one half b). negative one half c). -2 d).2
Answer:
d).2
Step-by-step explanation:
Use the following equation:
slope (m) = (y₂ - y₁)/(x₂ - x₁)
Let:
(x₁ , y₁) = (1 , -3)
(x₂ , y₂) = (2 , -1)
Plug in the corresponding numbers to the corresponding variables:
m = (-1 - (-3))/(2 - 1)
Simplify. Combine like terms. Remember to follow PEMDAS. Remember that two negatives = one positive:
m = (-1 + 3)/(2 - 1)
m = (2)/(1)
m = 2
d).2 is your slope.
~
Answer:
[tex]\displaystyle =2[/tex]
Step-by-step explanation:
The slope formula is ⇒ [tex]\displaystyle\frac{Y_2-Y_1}{X_2-X_1}[/tex]
[tex]\displaystyle Y_2=(-1)\\\displaystyle Y_1=(-3)\\\displaystyle X_2=2\\\displaystyle X_1=1\\[/tex]
[tex]\displaystyle \frac{(-1)-(-3)}{2-1}=\frac{2}{1}=2[/tex]
Therefore, the slope is 2, and the correct answer is 2.
Hope this helps!
Which are the solutions of x^2 = -13x – 4?
Using the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) to find the solutions, which are (13 + √185) / 2 and (13 - √185) / 2.
Explanation:To find the solutions of the equation x^2 = -13x - 4, we can rearrange it to the form ax^2 + bx + c = 0.
In this case, we have a = 1, b = -13, and c = -4.
We can then use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) to find the solutions.
Substituting the values into the formula, we get x = (-(-13) ± √((-13)^2 - 4(1)(-4))) / (2(1)). Simplifying further, we have x = (13 ± √(169 + 16)) / 2. T
herefore, the solutions of the equation are x = (13 + √185) / 2 and x = (13 - √185) / 2.
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A diet is to include at least 140 milligrams of Vitamin A and at least 145 milligrams of Vitamin B. These requirements can be obtained from two types of food. Type X contains 10 milligrams of Vitamin A and 20 milligrams of Vitamin B per pound. Type Y contains 30 milligrams of Vitamin A and 15 milligrams of Vitamin B per pound. If type X food costs $12 per pound and type Y food costs $8 per pound how many pounds of each type of food should be purchased to satisfy the requirements at the minimum cost? Round to the nearest hundredths.
To minimize the cost of the diet while meeting the vitamin requirements, we can use a system of linear equations to solve a linear programming problem. The optimal solution will provide the pounds of each type of food to be purchased.
Explanation:To solve this problem, we can use a system of linear equations. Let's assume that we buy x pounds of type X food and y pounds of type Y food. The requirements for Vitamin A and Vitamin B can be expressed as the following inequalities: 10x + 30y ≥ 140 (for Vitamin A) and 20x + 15y ≥ 145 (for Vitamin B). We also need to minimize the cost, which can be expressed as the objective function: Cost = 12x + 8y. So, we have a linear programming problem.
To find the minimum cost, we can graph the feasible region defined by the inequalities and find the corner point with the lowest cost. Alternatively, we can use a method like the Simplex algorithm to solve the system of equations and find the optimal solution. The solution will give us the values of x and y that satisfy the requirements and minimize the cost.
Once we find the optimal solution, we can round the values of x and y to the nearest hundredths and provide the student with the pounds of each type of food to be purchased.
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Which of the following graphs represents the translation of Rectangle ABDC over the following: (x,y) ---> (x+1, y-2) ???
I WILL MARK BRAINLIEST
please answer I'll do anything. I know I have the answer right, i just need to know how to show the work
Answer:
B.
Step-by-step explanation:
The spaces on the graph are x=3 nd y=7 so x+1= Move it up 1 space so it's at -4. And move y up 2 so it's at 3. That's why B is correct.
(I can't really be more specific, sorry.)
Divide 1,600 by 84, and then divide the answer by 3
Answer:
6.3
Step-by-step explanation:
1,600 / 84 = 19.04
19.04 / 3 = 6.3
To answer the student's question, we first divide 1,600 by 84 which gives around 19.0476, and then divide this result by 3, which gives the final answer of 6.3492.
Explanation:To solve this problem, we first need to divide 1,600 by 84. That will give us the first result. Next, we must take this first result and divide it by 3.
Let's take it step by step:
1,600 ÷ 84 ≈ 19.0476 (You can take it to the nearest 4 decimal points for simplicity). Now, take this result (19.0476) and divide it by 3. The answer is around 6.3492.
So after you divide 1,600 by 84, and then divide the answer by 3, the final result should be around 6.3492.
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Solve the inequality 2x2 + 10x < –8
Answer:
-4<x<-1
Step-by-step explanation:
To solve the problem, we divide the whole expression by 2:
2x^2 + 10x < –8 → x^2 + 5x < –4
→ x^2 + 5x + 4 < 0
Factorizing
→ (x+4)(x+1) < 0
The expression is ONLY negative when:
x>-4 and x<-1
Therefore, the solution is:
-4<x<-1
The inequality 2x2 + 10x < -8 should be rearranged to 2x2 + 10x + 8 < 0. It can't be factored so we must use the quadratic formula to solve it. The solution involves finding the values of x that make the function positive or negative.
Explanation:To solve the inequality 2x2 + 10x < -8, we first need to arrange the terms. To do so, we can subtract 8 from each side of the inequality obtaining 2x2 + 10x + 8 < 0.
However, this is a quadratic inequality that is best solved by factoring, if possible. Let's try to factor our quadratic expression, but keep in mind that not all quadratic expressions can be factored. In this case, it is not factorable. Therefore, it must be solved by using the quadratic formula, which is x = [-b ± sqrt(b2-4ac)]/2a.
Take note the a, b, and c values from our inequality (a=2, b=10, c=8), and substitute these values into the quadratic formula. However, this approach will solve for 'x' in an equation, not an inequality.
To solve for 'x' in the inequality, we have to find the values of 'x' that make the function positive or negative, depending on the type of the inequality. We determine those values by solving the equation f(x) = 0, where f(x) is the left side of our inequality.
It is a complex process that requires understanding of both quadratic functions and inequalities. Following these steps and calculating correctly should lead you to the solution.
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how do i solve x+3y=7
Answer:
If you're trying to find x or y, you cannot find the exact value since there is one equation. If there was, for example, 2 equations, [a system of equations] you could solve that. But here we only have one equation.
So, if you're trying to solve for x;
x = 7 - 3y
And if you're trying to solve for y;
y = 7/3 - x
The graph of y= -4x + 7 is:
Answer:
Your y-intercept is at (0,7). To plot other points, use your slope: -4. (go down 4, go right 1 OR go up 4 and left 1)
Step-by-step explanation:
Answer:
Graph is attached below
Step-by-step explanation:
[tex]y= -4x + 7[/tex]
To graph this linear function , we make a table
Assume some random number for x and find out y. Assume some positive and negative numbers for x
x y=-4x+7
-1 -4(-1)+7=11
0 -4(0)+7=7
1 -4(1)+7=3
The points we got are (-1,11) (0,7) (1, 3)
Plot all the points and join all the points by a line
The graph is attached below