Answer:
[tex]\large\boxed{-\dfrac{1}{3}-\dfrac{\sqrt{14}}{3}i,\ -\dfrac{1}{3}+\dfrac{\sqrt{14}}{3}i}[/tex]
Step-by-step explanation:
Use
[tex]ax^2+bx+c=0\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
[tex]3x^2+2x+5=0\\\\a=3,\ b=2,\ c=5\\\\b^2-4ac=2^2-4(3)(5)=4-60=-56\\\\\sqrt{b^2-4ac}=\sqrt{-56}=\sqrt{(4)(-14)}=\sqrt4\cdot\sqrt{-14}=2\sqrt{-14}[/tex]
Use
[tex]i=\sqrt{-1}[/tex]
[tex]\sqrt{-14}=\sqrt{(-1)(14)}=\sqrt{-1}\cdot\sqrt{14}=i\sqrt{14}[/tex]
Therefore:
[tex]x=\dfrac{-2\pm 2i\sqrt{14}}{2(3)}=-\dfrac{2}{6}\pm\dfrac{2i\sqrt{14}}{6}=-\dfrac{1}{3}\pm \dfrac{\sqrt{14}}{3}i[/tex]
The complex solution of the given quadratic equation 3x^2+2x+5=0 is x = (1/6)[ -2 ± i√56].
What is a quadratic equation?An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, and a must not be zero.
For example, 3x² + 6x + 8 = 0 here x has the highest term as 2 and the coefficient of x² is not zero.
As per the given quadratic equation, 3x^2+2x+5=0
x = [-2 ±√(4 - 4 x 3 x 5)]/(2 x 3)
x = (1/6)[ -2 ± i√56]
Hence "The complex solutions of the given quadratic equation 3x^2+2x+5=0 is x = (1/6)[ -2 ± i√56]".
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Find the simple interest paid one $600 is borrowed for three years at 6% per
Answer:
Principal amount= $600
Time= 3 years
Rate of Interest = 6%
SI= PRT/100
= 600x6x3/100
= $108
Interest is $108
Step-by-step explanation:
15 POINTS
Using Euler's Formula: F+V=E+2
A geometric solid has thirty-two faces and ninety edges. Find the number of vertices.
SHOW WORK
Answer:
V -E +F = 2
Vertices = 2 + Edges -Faces
Vertices = 2 + 90 -32
Vertices = 60
Step-by-step explanation:
9 pounds of ground beef cost $40.32. What is the price per ounce?
The price per ounce of ground beef, when 9 pounds cost $40.32, is $0.28.
Calculating Price per Ounce of Ground Beef
To calculate the price per ounce for ground beef when given the price for 9 pounds, we will first need to know how many ounces are in 9 pounds. Since there are 16 ounces in 1 pound, 9 pounds is equivalent to 144 ounces (9 pounds × 16 ounces/pound).
Next, we divide the total cost of the 9 pounds of ground beef by the total number of ounces to find the price per ounce. The total cost given is $40.32.
Price per ounce = Total cost \/ Total number of ounces
Price per ounce = $40.32 \/ 144 ounces
Price per ounce = $0.28
Therefore, the price per ounce of ground beef is $0.28.
The perimeter of a rectangular herb garden is 74 feet. If the length is 5 feet longer then 3 times it’s width what is the gardens width?
Answer:
width = 8 feet and length = 29 feet
Step-by-step explanation:
perimeter of a rectangle P = 2(L + W)
width W = x
length L = 3x + 5
P = 74 feet
74 = 2(3x + 5 + x)
37 = 3x + 5 + x
37 = 4x + 5
4x = 37 -5
4x = 32
x = 8
W = 8 feet
If f(x) = 5x - 2 and g(x) = 2x + 1, find (f + g)(x)
Answer:
(f + g)(x) = 7x - 1
Step-by-step explanation:
Given : f(x) = 5x – 2 and g(x) = 2x + 1
We have to find (f + g)(x)
Consider (f + g)(x) = f(x) + g(x)
Also, given f(x) = 5x – 2 and g(x) = 2x + 1
Substitute, we have,
f(x) + g(x) = 5x - 2 + 2x + 1
Like Terms are terms having same variable with same degree.
Simplify by adding like terms, we have,
f(x) + g(x) = 5x + 2x - 2 + 1
f(x) + g(x) = 7x - 1
Thus, (f + g)(x) = 7x - 1
Answer:
Thus, (f + g)(x) = 7x - 1
Step-by-step explanation:
(f + g)(x) = 7x - 1
Step-by-step explanation:
Given : f(x) = 5x – 2 and g(x) = 2x + 1
We have to find (f + g)(x)
Consider (f + g)(x) = f(x) + g(x)
Also, given f(x) = 5x – 2 and g(x) = 2x + 1
Substitute, we have,
f(x) + g(x) = 5x - 2 + 2x + 1
LIKE TERMS are terms having same variable with same degree.
Simplify by adding like terms, we have,
f(x) + g(x) = 5x + 2x - 2 + 1
f(x) + g(x) = 7x - 1
Thus, (f + g)(x) = 7x - 1
Barry is 8 years older than his sister. In 3 years, he will be twice as old as she will be then. How old is each now?
Final answer:
Barry is currently 13 years old and his sister is 5 years old. This is determined by setting up a system of equations based on the given information and solving for their current ages.
Explanation:
The question involves solving a system of equations to find the current ages of Barry and his sister. Let's let B represent Barry's current age and S represent his sister's current age. According to the problem, Barry is 8 years older than his sister (B = S + 8). In 3 years, Barry will be twice as old as his sister will be then (B + 3 = 2(S + 3)).
Write down the two equations: B = S + 8 and B + 3 = 2(S + 3).
Substitute B from the first equation into the second equation: (S + 8) + 3 = 2(S + 3).
Simplify and solve for S: S + 11 = 2S + 6.
Rearrange the equation to find S: S = 5.
Substitute the value of S back into the first equation to find B: B = 5 + 8, so B = 13.
Therefore, Barry is currently 13 years old and his sister is 5 years old.
Multiply and simplify.
(x + 3)(x - 8)
A) 2x - 24
B) x2 - 5x + 24
C) x2 - 5x - 24
D) x2 - 8x - 24
Answer:
x^2-5x-24
Step-by-step explanation:
Final answer:
The product of the binomials (x + 3)(x - 8) is simplified using the FOIL method to get x² - 5x - 24. We multiply each term in the first parenthesis by each term in the second and combine like terms.
Explanation:
To multiply and simplify the expression (x + 3)(x - 8), we will use the FOIL method (First, Outer, Inner, Last). This method involves multiplying each term in the first parenthesis by each term in the second parenthesis.
First: x times x = x²
Outer: x times -8 = -8x
Inner: 3 times x = +3x
Last: 3 times -8 = -24
Next, combine like terms (-8x and +3x):
x² - 8x + 3x - 24 = x² - 5x - 24
The simplified form of the given expression is x² - 5x - 24, which corresponds to option C).
Tania analyzed the relationship between student test scores and the number of hours studied. She calculated the trend line to be y = 6.8x + 60, where x is the number of hours studied and y is the score. Which closet is to the score for a student who studied 3 hours?
A 80
B 85
C 90
D 95
Answer:
A: 80
Step-by-step explanation:
As we know that the trend line is [tex]y=6.8x+60[/tex], we can plug in x=3 in order to find out the estimated score for 3 hours of studying
[tex]y=6.8x+60\\\\y=6.8(3)+60\\\\y=20.4+60\\\\y=80.4[/tex]
y=80.4 is closest to A: 80
Answer:
Option A, 80
Step-by-step explanation:
Trend line is given by y = 6.8x + 60
where x = number of hours she studied
y = score
Now we have to calculate the score for a student for studied for 3 hours.
y = 6.8 × 3 + 60
= 20.4 + 60
= 80.40
≈ 80
Option A will be the answer.
Perform the indecated operation. 5 over 9 times 3 over 7
Answer:
[tex]\frac{5}{21}[/tex]
Step-by-step explanation:
[tex]\frac{5}{9}[/tex] × [tex]\frac{3}{7}[/tex] = [tex]\frac{15}{63}[/tex] = [tex]\frac{5}{21}[/tex]
Answer:
5/63
Step-by-step explanation:
that would be:
5 3 5
----- * ------ = (after reduction) ----------- = 5/63
9 7 3(21)
The length of a rectangle is 2 cm more than four times the width. If the perimeter of the rectangle is 84 cm, what are its dimensions
Answer:
Length 34 cm
Width 8 cm
Step-by-step explanation:
Let
x-----> the length of the rectangle
y----> the width of the rectangle
we know that
The perimeter of rectangle is equal to
P=2(x+y)
P=84 cm
so
84=2(x+y)
42=x+y ----> equation A
x=4y+2 -----> equation B
substitute equation B in equation A and solve for y
42=(4y+2)+y
5y=42-2
y=40/5=8 cm
Find the value of x
x=4y+2 ----> x=4(8)+2=34 cm
The dimensions are
Length 34 cm
Width 8 cm
The width of the rectangle is 8 cm and the length is 34 cm.
To determine the dimensions of the rectangle, we need to set up equations based on the given information.
Let's denote the width of the rectangle by W (in cm). According to the problem, the length L is 2 cm more than four times the width. Therefore, we can express the length as:
L = 4W + 2
The perimeter of a rectangle is calculated using the formula:
Perimeter = 2L + 2W
Given that the perimeter is 84 cm, we can substitute the expressions for L and W into the perimeter formula:
84 = 2(4W + 2) + 2W
Simplify the equation:
84 = 8W + 4 + 2W84 = 10W + 4Subtract 4 from both sides: 80 = 10WDivide both sides by 10: W = 8Now that we have the width, we can find the length:
L = 4(8) + 2 = 32 + 2 = 34
Thus, the dimensions of the rectangle are:
Width: 8 cmLength: 34 cm.Determine whether the piece of paper can be used to wrap the rectangular prism exactly once without any gaps or overlaps.
Answer:
No, because they do not have the same area.
Step-by-step explanation:
For the piece of paper to cover once without overlap or gaps, it must have the same area as the the total surface area of the prism.
The total surface area of the rectangular prism is: [tex]2(3.5\times2+3.5\times1.5+2\times1.5)=30\frac{1}{2}in^2[/tex]
The area of the rectangular sheet is:[tex]5\times 8=40in^2[/tex]
Since the area of the piece of paper is more than the surface area of the rectangular prism there will be an overlap.
Find the mode of this set of data 18,27,28,44,44,50,67
Answer:
44
Step-by-step explanation:
the mode of a set of data is the value that occurs most often, in this case it is 44
44 would be the mode of the data.
The mode of the data is the number that occurs the most. It is always best to put the numbers in order from least to greatest so we can see what number appears more than once.
18,27,28,44,44,50,67
When we look at this set of numbers, we can see that 44 appears more than any other number. Therefore, the mode of these numbers is 44.
In triangle JKL, m of angle J = 90, m of angle K = 30, and m of angle L = 60. Which of the following statements about triangle JKL are true? Check all that apply.
A) KL= root 3 (JL)
B) KL= 2(JL)
C) JK= root 3 (JL)
D) JK= 2(JL)
E) JL= root 3/2 (KL)
F) JK= root 3/2 (KL)
Answer:
# Answer B is true ⇒ KL = 2(JL)
# Answer C is true ⇒ JK = √3(JL)
# Answer F is true ⇒ JK = √3/2(KL)
Step-by-step explanation:
* Lets explain the ratio between the sides of the triangle
- In Δ JKL
∵ The measure of angle J is 90°
∴ KL is the hypotenuse
∵ The measure of angle K is 30°
∴ JL is the opposite side to angle 30°
∵ The measure of angle L is 60°
∴ JK is the opposite side to the angle 60°
- There is a fact in the triangle which has angles 30° , 60° , 90°
# The length of the side opposite to the angle of measure 30° is
half the length of the hypotenuse
∵ KL is the hypotenuse
∵ JL is the opposite side to the angle of measure 30°
∴ JL = 1/2 KL OR KL = 2 JL
# The length of the side opposite to the angle of measure 60° is
√3 the length of the opposite side to the angle 30°
∵ JK is the opposite side to the angle of measure 60°
∵ JL is the opposite side to the angle of measure 30°
∴ JK = √3 JL
# The length of the side opposite to the angle of measure 60° is also
half √3 the length of the hypotenuse
∵ JK is the opposite side to the angle of measure 60°
∵ KL is the hypotenuse
∴ JK = √3/2 KL
- From the relation above
# Answer B is true ⇒ KL = 2(JL)
# Answer C is true ⇒ JK = √3(JL)
# Answer F is true ⇒ JK = √3/2(KL)
Answer:
It is a and B in C & D
Step-by-step explanation:
These are the closest answers
If 5 x 6 = 20 + y, what is the value of y?
5*6=20+y
30=20+y
30-20=20-20+y
10=y
Check answer by using substitution method
5*6=20+y
30=20+10
30=30
Answer is y=10
5 x 6 = 30
y + 20 = 30
y + 20 - 20 = 30 - 20
y = 10
ASAP RIGHT NOW NEW FOR ME
Answer:8m+20
Step-by-step explanation:
U have too add the perimeter than multiple area
If I make 180 dollars a week. How many weeks does it take to reach 7,500 dollars?
Answer:
it would take you 41 weeks
(rounded to 42 if that is a choice answer because when you divide 7500 by 180 it gives you 41.66666)
Step-by-step explanation:
A right rectangular prism has a length of 5 centimeters, a width of 8 centimeters, and a height of 4
centimeters.
What is the volume of the prism?
Enter your answer in the box.
cm3
Answer:
160cm^3
Step-by-step explanation:
V=Lwh
so, (5x8x4)
v=160cm^3
Hope my answer has helped you!
A flat screen television has a 50 inch diagonal and a height of 25 inches. How wide is the television rounded to the nearest tenth?
(Use the Pythagorean Theorem)
Answer
inches
Answer:
The wide of television is [tex]43.3\ in[/tex]
Step-by-step explanation:
Let
x----> the wide of television
we know that
Applying the Pythagoras Theorem
[tex]50^{2}=x^{2}+25^{2}[/tex]
Solve for x
[tex]x^{2}=50^{2}-25^{2}[/tex]
[tex]x^{2}=1,875[/tex]
[tex]x=43.3\ in[/tex]
The sum of two integers is -8.if one integer is 12 then the other is?
Answer:
2 and -10
Step-by-step explanation:
let the two integers be x and y.
so, x+y=-8---‐-------(1)
and,
x=y+12-------‐-----(2)
from (1) and (2),
(y+12)+y=-8
or, 2y=-8-12
or, y=-20/2
or, y=-10
substitutting the value of y in (2),
x=-10+12=2
therefore the required integers are 2 and -10.
Answer:
-20
Step-by-step explanation:
→ The sum of two integers is -8.
⇒ One will be called "x" and the other will be called "y"
→ One integer is 12 ( "y" )
Let's set up an equation x = ? and y = 12 so,
→ x + y = -8
( Substitute y for 12 )
→ x + 12 = -8
( -12 from both sides to isolate x )
→ x = -20
Fernando bought a new watch set that includes 5 watch faves and 7 colored bands . From how many different watches can he choose?
Answer:
35 different watches
Step-by-step explanation:
I don't know if this is correct so sorry if it's not.
5 faces x 7 bands.
So, 5 x 7
=
35
i hope that helps... it might not, sorry
Answer:
35 different watches/combos.
Step-by-step explanation:
5 watches, 7 different wrist bands
5 x 7 = 35
1 watch consist of 1 watch and 1 band so he can have 35 different watch combinations.
if f(x) and f^-1(x) are inverse functions of each other adn f(x) - 2x+5, what is f^-1(8)?
Answer:
[tex]f^{-1}(8)=1.5[/tex]
Step-by-step explanation:
If y=f(x)=2x+5, to find the inverse function express x n terms of y:
[tex]y=2x+5\\ \\2x=y-5\\ \\x=\dfrac{y-5}{2}[/tex]
Now, change y into x and x into y:
[tex]y=\dfrac{x-5}{2}\\ \\f^{-1}(x)=\dfrac{x-5}{2}[/tex]
Substitute x=8 into [tex]f^{-1}(x)[/tex] expression:
[tex]f^{-1}(8)=\dfrac{8-5}{2}\\ \\f^{-1}(8)=\dfrac{3}{2}=1.5[/tex]
Answer:
3/2
Step-by-step explanation:
f(x) = 2x+5
f⁻¹(2x+5) = x
2x+5 = 8 => 2x+5 = 8 => 2x = 3 =>
=> x = 3/2
=> f⁻¹(8) = 3/2
what is the other binomial?
Answer:
11x² -3x
Step-by-step explanation:
We have the product of a sum, of two binomials and one of the initial binomials but we'd like to know the other one.
It's very much like A + B = C, and we have both B and C, but looking to find A.
What would you do in the ABC scenario?
You'd isolate A by transforming the equation to: A = C - B
In this case, C = 12x² - 5x and B = x² - 2x...
A = (12x² - 5x) - (x² - 2x) = 12x² - 5x - x² + 2x
A = 11x² -3x
Evaluate the dot product of (2,4) and (1,2)
ANSWER
The dot product is 10.
EXPLANATION
The given vectors are (2,4) and (1,2).
If we have the vectors
u=(a,b) and v=(c,d)
Then the dot product of the two vectors is given by
[tex]u \bullet \: v = ac + bd[/tex]
This implies that,
[tex](2,4) \bullet(1,2) = 2 \times 1 + 4 \times 2[/tex]
This simplifies to;
[tex](2,4) \bullet(1,2) = 2 + 8 = 10[/tex]
Answer:
-6
Step-by-step explanation:
took test
Solve for X in this equation
Answer: x=e
Step-by-step explanation:
x= ln(e^e)
using ln(e^x)= x x ln (e), transform the expression
x= e x ln (e)
the natural logatithm of e equals 1
x=e x 1
any expression multiplied by 1 remains the same
x=e alternative form: x= 2.71828
ANSWER
[tex]x = e[/tex]
EXPLANATION
The given logarithmic equation
[tex]x = ln_{e}(e) [/tex]
Note that
[tex] ln(e) = 1[/tex]
Using the power rule of logarithms,
[tex]x =e \ln(e)[/tex]
This gives us:
[tex]x =e (1)[/tex]
Simplify to get:
[tex]x = e[/tex]
a Potter uses 3/5 of a pound of clay to make a bowl how many bowls could the Potter make from 10 lb of clay
Answer:
16
Step-by-step explanation:
Let's say x is the number of bowls. If each bowl needs 3/5 pound of clay, then the total clay needed is 3/5 * x. We know the total amount is 10 pounds. Therefore:
10 = 3/5 x
Divide:
x = 10 / (3/5)
To divide by a fraction, we need to multiply by the reciprocal:
x = 10 * (5/3)
x = 50/3
x = 16 ⅔
We can't have part of a bowl, so we must round down. The potter can make 16 bowls.
Which of the following digits could replace the □ in the tens place to make this statement true? 88,5□1 rounds to 88,500 if we round to the nearest hundred.
chose all answers that apply:
:0
:2
:4
To make 88,5□1 round down to 88,500, the digit that replaces □ must be less than 5; therefore, the replacements could be :0, :2, or :4.
potential
Explanation:The given number is '88,5□1' and we want to find the digit that replaces the □ and rounds the number to '88,500' when we round to the nearest hundred. An important point to remember about rounding is that if the digit in the tens place is less than 5, we round down, while if it's 5 or above, we round up. Since we want to round down to 88,500, the digit that replaces □ in the tens place must therefore be less than 5. So, the possible digits could be :0, :2, or :4.
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7cm
10cm
4cm
The area of the compound shape is 106cm.
work out the size of x.
Answer: 9cm
Step-by-step explanation:
[tex]7x10=70cmx^{2}[/tex]
4(x)=4x
1.) 70+4x=106 (subtract 70 on both sides)
2.)4x=36 (divide 4 on both sides)
3.)x=9
answer=9
7x10=70cmxsquare
4(x)=4x
70+4x=106
4x=36
x=36/4
x=9
Driving across country, mike drove for 3 hours at a certain rate and for 4 hours at a rate 10 mph faster. If his total distance covered was 495 miles, then what was his slower rate?
Mike's slower rate of driving across country was 65 mph.
Explanation:Let the slower rate be represented as r mph. The faster rate will be r+10 mph. The total distance covered (495 miles) is the product of the time and the rate:
The distance covered at the slower rate is 3r miles.The distance covered at the faster rate is 4(r+10) miles.Adding up the distances, we get:
3r + 4(r+10) = 495Simplifying the equation:
3r + 4r + 40 = 495Combining like terms:
7r + 40 = 495Subtracting 40 from both sides:
7r = 455Dividing by 7:
r = 65Therefore, the slower rate is 65 mph.
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The correct answer for the slower rate at which Mike drove is 65 mph.
To find the slower rate at which Mike drove, let's denote the slower rate as r mph. Mike drove at this rate for 3 hours. He then drove for 4 hours at a rate that is 10 mph faster than the slower rate, which would be r + 10 mph.
The total distance covered by Mike is the sum of the distances covered at each rate. We can set up an equation to represent this:
[tex]\[ 3r + 4(r + 10) = 495 \][/tex]
Expanding the equation, we get:
[tex]\[ 3r + 4r + 40 = 495 \][/tex]
Combining like terms, we have:
[tex]\[ 7r + 40 = 495 \][/tex]
Now, we need to solve for r We can do this by subtracting 40 from both sides of the equation:
[tex]\[ 7r = 495 - 40 \] \[ 7r = 455 \][/tex]
Dividing both sides by 7 to solve for r, we get:
[tex]\[ r = \frac{455}{7} \] \[ r = 65 \][/tex]
Therefore, the slower rate at which Mike drove is 65 mph.
Find negative square root of 36. ±6 −6 6 18
Answer:
The answer is -6
Step-by-step explanation:
Hope it helped you!
Answer:
B. [tex]-6[/tex]
Step-by-step explanation:
We are asked to find the negative square root of 36.
We can write our given problem as [tex]-\sqrt{36}[/tex].
We can write 36 as 6 to second power.
[tex]-\sqrt{36}=-\sqrt{6^2}[/tex]
Using radical rule [tex]\sqrt[n]{a^n}=a[/tex], we will get:
[tex]-\sqrt{6^2}=-6[/tex]
Therefore, the negative square root of 36 is negative six and option B is the correct choice.
See attached question below
ANSWER
A. 10,000
EXPLANATION
The given rational function is
[tex]f(x) = \frac{1}{x - 2} [/tex]
This function is not defined at x=2.
But as we are picking x-values that are closer and closer to 2, the functional values grows bigger and bigger positively or negatively without bounds.
Therefore a possible value of f(x) as x is close to 2 is 10,000.
The correct answer is A.