Answer:
x = -2
Step-by-step explanation:
We are given the logarithmic base 2 equation of:
[tex]\displaystyle{\log_2 (4x) + \log_2 (x+1) = 3}[/tex]
Apply logarithm property of addition where:
[tex]\displaystyle{\log_a M + \log_a N = \log_a MN}[/tex]
Therefore, we will write new equation as:
[tex]\displaystyle{\log_2 [4x(x+1)] = 3}[/tex]
Apply logarithm to exponential form using:
[tex]\displaystyle{\log_a M = N \to a^N = M}[/tex]
Thus, another new rewritten equation is:
[tex]\displaystyle{2^3 = 4x(x+1)}\\\\\displaystyle{8 = 4x(x+1)}\\\\\displaystyle{2=x(x+1)}[/tex]
Expand the expression in and arrange the terms in quadratic expression:
[tex]\displaystyle{2=x^2+x}\\\\\displaystyle{0=x^2+x-2}\\\\\displaystyle{x^2+x-2=0}[/tex]
Solve for x:
[tex]\displaystyle{(x+2)(x-1)=0}\\\\\displaystyle{x=-2,1}[/tex]
These are potential solutions to the equation. To find extraneous solution, you’ll have to know the domain of logarithm function. We know that logarithm’s domain is defined to be greater than 0. Henceforth, anti-logarithm must be greater than 0.
( 1 ) 4x > 0, x > 0
( 2 ) x + 1 > 0, x > -1
Therefore, our anti-log must be greater than 0, so any solutions that are equal or less than 0 will be considered as extraneous solution.
Hence, x = -2 is the extraneous solution.
What is f(x) = 8x² + 4x written in vertex form?
F(x) = 8x + - 1.
f(x) = 8√(x + 1)² - 1/6
f(x) = 8√x+¹²-2
f(x) = 8(x + 1)² - 4
[tex]8x^{2}+4x\\\\=8\left(x^{2}+\frac{1}{2}x \right)\\\\=8\left(x+\frac{1}{4} \right)^{2}-8\left(\frac{1}{16} \right)\\\\\boxed{f(x)=8\left(x+\frac{1}{4} \right)^{2}-\frac{1}{2}}[/tex]
In the figure above ac bisects bcd. what is the length of cd?
In the the figure above AC bisects BCD in the quadrilateral, the length of CD will be 6cm.
How to calculate the value?It should be noted that the longer diagonal is a perpendicular bisector of the shorter diagonal.
From the diagram given, the value of AD is 6cm. Therefore, according to the angle bisector theorem, the value of CD is 6cm as they're congruent.
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solve for x -
[tex]\bold{x {}^{2} - 25 + 16 = 0}[/tex]
ty! ~
Hey!
Your equation is -
[tex]x {}^{2} - 25 + 16 = 0[/tex]
[tex]x {}^{2} - 9 = 0[/tex]
We can write 9 as 3² (Why?) because it forms identity a² - b² = (a + b) (a - b) .
[tex]x {}^{2} - {3}^{2} = 0[/tex]
[tex](x - 3)(x + 3) = 0[/tex]
This gives the information that either,
x - 3 = 0 or x + 3 = 0
So, x = 3 or x = -3
Hope it helps, you may ask if you have any confusions ^-^
Answer:
x = 3, x = -3
Step-by-step explanation:
x² - 25 + 16 = 0
⇒ x² - 9 = 0
• Add 9 to both sides:
⇒ x² = 9
• Square root both sides of equation:
x = [tex]\sqrt{9}[/tex]
x = ± 3
x = 3 and x = -3
The position of an object at time t is given by s(t) = 1 - 12t. Find the instantaneous velocity at t = 2 by finding the derivative.
The change of displacement with respect to time is defined as speed. Velocity is a vector quantity. The instantaneous velocity at t = 2 is -12.
What is velocity?The change of displacement with respect to time is defined as Velocity. velocity is a vector quantity. It is a time-based component. Its standard unit is m/sec.
Given that the position of an object at time t is given by s(t) = 1 - 12t. Therefore, the velocity of the object can be written as,
V = ds/dt
= d(1 - 12t)/dt
= -12
s'(t) = -12
Now, the instantaneous velocity at t=2 is,
s'(2) = -12
Hence, the instantaneous velocity at t = 2 is -12.
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Hi- I need help please to get my HS diploma...did not graduate :(
Find the remainder when f(x)=x3−6x2+3x−1 is divided by 2x−3.
The remainder from the division of the algebraic equation is -53/8.
What is the remainder of the algebraic expression?The remainder of the algebraic expression can be determined by using the long division method.
Given that:
[tex]\mathbf{f(x) = \dfrac{x^3 - 6x^2 + 3x - 1}{2x-3}}[/tex]
where:
The divisor = 2x -3Using the long division method, we have:
[tex]\mathbf{= \dfrac{x^2}{2} +\dfrac{-\dfrac{9x^2}{2}+3x -1 }{2x-3}}[/tex]
[tex]\mathbf{= \dfrac{x^2}{2}-\dfrac{9x}{4} +\dfrac{-\dfrac{-15x}{4}-1 }{2x-3}}[/tex]
[tex]\mathbf{= \dfrac{x^2}{2}-\dfrac{9x}{4} -\dfrac{15}{8}+\dfrac{-\dfrac{53}{8} }{2x-3}}[/tex]
[tex]\mathbf{= \dfrac{x^2}{2}-\dfrac{9x}{4} -\dfrac{15}{8}-\dfrac{53 }{8(2x-3)}}[/tex]
Therefore, we can conclude that the remainder is -53/8.
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Find the area of the following figure:
The area of the figure is 11. 27 cm²
How to find the areaThe figure given is a parallelogram
Note that the formula for area of a parallelogram is given as;
Area = base × height
Base = 2. 3cm
height = 4. 9cm
Area = 2. 3 × 4. 9
Area = 11. 27 cm²
Thus, the area of the figure is 11. 27 cm²
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A retail store in Des Moines, Iowa, receives shipments of a particular product from Kansas City and Minneapolis. Let x = number of units of the product received from Kansas City y = number of units of the product received from Minneapolis a. Write an expression for the total number of units of the product received by the retail store in Des Moines. b. Shipments from Kansas City cost $0.20 per unit, and shipments from Minneapolis cost $0.25 per unit. Develop an objective function representing the total cost of shipments to Des Moines. c. Assuming the monthly demand at the retail store is 5000 units, develop a constraint that requires 5000 units to be shipped to Des Moines. d. No more than 4000 units can be shipped from Kansas City, and no more than 3000 units can be shipped from Minneapolis in a month. Develop constraints to model this situation. e. Of course, negative amounts cannot be shipped. Combine the objective function and constraints developed to state a mathematical model for satisfying the demand at the Des Moines retail store at minimum cost. Solve using linear programming
The expression for the total number of units of the product received is x + y.
How to depict the information?The total will be =y+x because it will be the sum of the two shipment quantities.
b. Cost = $.20(x)+$.25(y) because the cost will be the sum of the costs per unit for each shipment.
c. 5000 =x+y allows for any combination of x and y units.
d. Total will be: x+y where x<=4000 and y<=3000
e. The minimum cost is where a and b are minimized in demand =ax+by. We cannot determine a maximum value for a or b because we do not know what the demand is. If the problem calls for referral to part b, then use the equation demand=$.20x
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What is this math problem in Simply form 3-4(-2-7)
1. make sure you understand what the problem really is
3-4(-2-7)
= 3 + -4 × (-2 + -7)
2. always begin with the brakes
3 + -4 × (-2 + -7)
= 3 + -4 × (-9)
3. then ² and [tex] \sqrt{?} [/tex]
4. then × and and ÷
3 + -4 × (-9)
= 3 + 36
5. lastly + and -
3+36
=39
Jacob can take any one of three routes from school to the mall, and one of five possible routes from the mall to his house. If he doesn’t retrace his steps, how many different ways can Jacob walk from school to home?
Answer:
15 routes
Step-by-step explanation:
3 x 5 = 15 routes
I need the answer pls, any of y'all's answers would help!
Answer:
1
Step-by-step explanation:
The differences in the values on each spot on the line is 2, because -5 is in between -7 and -3. Going to the right means that the values are increasing. We go from -7, to -5, to -3, to -1, and finally, to 1.
Brainliest, please :)
A credit card company charges a 4% balance transfer fee. If you want to transfer a balance of $2,100 to this credit card, what fee would you pay?
Answer:
$84
Step-by-step explanation:
So since it charges 4%, this means you need to find 4% of what you're transferring. To find x% of a number, you need to convert the percentage to a decimal, and then multiply it by the number you're trying to find x% of. So in this case 4%/100 = 0.04. Now take this decimal and multiply it by 2100, and this will give you 84. So the fee would be $84
Answer:
$84
Step-by-step explanation:
= 4% × $2100
= 0.04 × $2100
= $84
Your original grades on your exams were: 75, 85, 90, 90, 100, 75, 80, 100, 90. You retake the exams and now your scores are: 85, 85, 100, 90, 100, 85, 80, 100, 90. By how many percentage points did you improve your average score?
Answer:
Your average improved by 13 [tex]\frac{1}{3}[/tex] %.
Step-by-step explanation:
First set:
(75 + 85 + 90 + 90 + 100 + 75 + 80 + 100 + 90) ÷ 9 = 77 [tex]\frac{2}{9\\}[/tex] %
Second set:
(85 + 85 + 100 + 90 + 100 + 85 + 80 + 100 + 90) ÷ 9 = 90 [tex]\frac{5}{9}[/tex] %
Improvement:
90 [tex]\frac{5}{9}[/tex] % - 77 [tex]\frac{2}{9\\}[/tex] % = 13 [tex]\frac{1}{3}[/tex] %
How do I calculate the 2nd question?
(Attached the file)
Answer:
0.29287
Step-by-step explanation:
You want to use the binomial cumulative distribution function (binomcdf). The purpose of this function is that it allows you to obtain the probability of observing less than or equal to x successes in n trials, with the probability p of success on a single trial.
x in this case is 182, n is 200, and p is 0.9005. Essentially by inputting these values, we find the probability that out of 200 passengers with a 0.9005 chance of getting on the plane, that 182 or less show up.
This gives us P(x <= 182) = 0.70713. (You need your calculator for this step, input the values above in the binomcdf function)
We want to find the probability of more than 182 passengers showing up though, so it would be 1 - 0.70713 = 0.29287.
Thus the probability that when 200 reservations are accepted for the flight, there are more passengers than seats available is 0.29287.
Please help!! Very much appreciated!! Consider the following set of real numbers. Which of the following lists all of the rational numbers in the set?
Answer:
The second choice.
Step-by-step explanation:
A rational number means that you are able to write the number as a fraction with integers in the numerator and the denominator. An integers are the whole numbers and their opposites ...-3,-2,-1,0,1,2,3...
The square root of 4 is 2. 2 can be written 2/1 (an integer over and integer)
-square root of 2 cannot be written as an integer over an integer. When this number is written as a decimal number it never repeats and it never terminates so it cannot be written as an integer over and integer.
-2/3 is written in the form of an integer over and integer.
1 can be written 1/1
9/8 is written as an integer over an integer.
The square root of 9/4 would be 3/2. (3/2)(3/2) = 9/4.
2.4 repeating can be written as an integer over an integer. Any decimal that repeats or terminates can be written as an integer over and integer. It actual happens to be 22/9.
The square root of 10 written as a decimal does not repeat or terminate so it is not rational. Only perfect squares are rational.
Which line has no slope?
a
O Line a
O Line b
O Line c
O Line d
Fifteen hungry kittens found a bucket of sausages. Each time
they cut a sausage or a part of a sausage, they cut it into exactly
two new pieces. They made 18 cuts in all and each kitten got 2
(maybe unequal) pieces of sausage. How many sausages were in
the bucket?
Answer:
2.4 SAUSAGES
Step-by-step explanation:
18 x 2 = 36
36/ 15 = 2.4
Work out 0.8 million - 1.76 thousand
The value of the given task 0.8 million - 1.76 thousand is 798,240.
Place value0.8 million
= 0.8 × 1,000,000
= 800,000
1.76 thousand
= 1.76 × 1000
= 1,760
So,
0.8 million - 1.76 thousand
= 800,000 - 1,760
= 798,240
Therefore, 0.8 million - 1.76 thousand is 798,240
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If IK=JK, find mlJ
A. 72°
B. 82°
C. 122°
D. 134°
Answer:
D.134
Step-by-step explanation:
In order to slove the following system of equations by addition, which of the following could you do before adding the equations so that one variable will be eliminated when you add them?
4x-2y=7
3x-3y=15
Answer:
B. use multipliers -3 and 2
Step-by-step explanation:
If you're solving the system by "addition," multiplying the equations by the chosen numbers needs to result in opposite coefficients for one of the variables.
Effect of answer choicesA. The system becomes ...
12x -6y = 2112x -12y = 60No variable has opposite coefficients.
__
B. The system becomes ...
-12x +6y = -216x -6y = 30The y-variable has opposite coefficients. This is a good choice.
__
C. The system becomes ...
12x -6y = 216x -6y = 30No variable has opposite coefficients.
__
D. The system becomes ...
4/3x -2/3y = 7/33x -3y = 15No variable has opposite coefficients.
__
Additional comment
The simplest "no brainer" solution is to use the coefficients of one of the variables, negating one of them. Multiply each equation by the opposite equation's coefficient. Here, the y-coefficients are -2 and -3, and the multipliers of answer choice B are -3 and 2, consistent with this advice. (The sign of 2 was changed.)
Answer:
B.
Step-by-step explanation:
Multiply the top equation by -3 and the bottom equation by 2
-12x+6y=-21
6x-6y= 30
-6x = 9 Variable "y" is removed
Hope this helps
If the sides of a polygon are x + 3 units, 2x – 4 units, 5x units, and 7x + 8 units, then find the perimeter of the polygon.
Answer:
The answer is 15x + 7
Step-by-step explanation:
Since the perimeter of a REGULAR polygon is the number of sides multiplied by the length, and the lengths of the sides are all different, it can be assumed that this is an irregular polygon. That means the formula to find the perimeter changes to all the sides added together.
1 x + 3
2 x - 4
5 x + 0
+ 7 x + 8
15 x + 7
DECIDE WHEATHER THE ORDER PAIR(-3,-5) IS A SOLITION OF THE EQUATION 7X-5Y=15
Answer:
(-3, -5) is not a solution of the equation 7x - 5y = 15
Step-by-step explanation:
1) Knowing that coordinates are given to us as (x, y), plug in the ordered pair into the equation. [tex]7(-3)-5(-5)=15[/tex]
2) Do PEMDAS on the left side of the equation
[tex]-21+25=15[/tex][tex]4=15[/tex][tex]4\neq 15[/tex]Answer:
Not a solution
Step-by-step explanation:
Hello!
We can determine if it's a solution by plugging in the x and y-values of the point into the equation. If both sides are equal to each other, then it is a solution.
-3 is the x-value and -5 is the y-value, since an ordered pari is written in the form of (x,y).
Evaluate7x - 5y = 157(-3) - 5(-5) = 15-21 + 25 = 154 ≠ 15Since both sides are not equal to each other, (-3,-5) is not a solution.
Which graph represents a function with direct variation? A coordinate plane with a line passing through (negative 4, 0) and (0, negative 2). A coordinate plane with a line passing through (negative 5, 4) and (0, 3). A coordinate plane with a line passing through (negative 4, negative 6) and (0, 3). A coordinate plane with a line passing through (negative 1, negative 4), (0, 0) and (1, 4).
A line passing through the points (-1,-4),(0,0) and (1,4).
The correct option is (B).
What is graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The only line that passes through origin is having coordinates (-1,-4),(0,0) and (1,4).
m= y/x
m = -4/-1
m=4
Hence, the linear equation is y=4x, which represent in direct variation.
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Answer:
C
Step-by-step explanation:
If the line is a direct variation it has to go through the origin
Which could be the entire interval over which the
function, f(x), is positive?
O (-∞, 1)
O (-2, 1)
O (-∞, 0)
O (1,4)
Answer:
(-2, 1)
Step-by-step explanation:
So assuming this table contains all the zeroes of f(x) and all of them having a multiplicity of 0. This quadratic equation can be defined as
[tex]f(x) = a(x+2)(x-1)[/tex]. So if you look at the top where its f(-3) then f(-2) then f(-1) it appears to be increasing and then somewhere in between f(-1) and f(0) it has a turning point in which it starts decreasing. and then after f(1) it no longer outputs positive values and appears to be going towards negative infinity. So the interval in which f(x) is positive seems to be finite and using this table it appears to be at (-2, 1) and make sure to use parenthesis since at f(-2) and f(1) it's not positive, because 0 is not positive, so it's not included in the interval
It would take 15men 8days to dig a trench of 240metres long.How many days would it take 18men to dig a trench of 360meters long working at the same Rate.
Using proportions, it is found that it would take 6.4 days for 8 men to dig a trench of 360 meters long working at the same rate.
What is a proportion?A proportion is a fraction of a total amount, and the measures are related using a rule of three.
In this problem, the rule of three is given as follows:
8 days - 15 men - 240 meters
x days - 18 men - 360 meters
The number of men is inverse proportional to the number of days, hence we divide 18 by 15 instead of 15 by 18, then:
[tex]\frac{8}{x} = \frac{18}{15} \times \frac{24}{36}[/tex]
[tex]\frac{8}{x} = \frac{6}{5} \times \frac{4}{6}[/tex]
[tex]\frac{8}{x} = \frac{5}{4}[/tex]
Applying cross multiplication:
5x = 32.
x = 32/5
x = 6.4 days.
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round the answer to the nearest
By knowing the blood pressure and applying the quadratic formula, the age of a man whose normal blood pressure is 129 mm Hg is 40 years old.
How to use quadratic equations to determine the age of a man in terms of blood pressure
In this problem we have a quadratic function that models the blood pressure as a function of age. As the latter is known, we must use the quadratic formula to determine the former:
129 = 0.006 · A² - 0.02 ·A + 120
0.006 · A² - 0.02 · A - 9 = 0
[tex]A = \frac{0.02 \pm \sqrt{0.006^{2}-4\cdot (0.006)\cdot (- 9)}}{2\cdot (0.006)}[/tex]
A = 1.667 + 38.733
A = 40
By knowing the blood pressure and applying the quadratic formula, the age of a man whose normal blood pressure is 129 mm Hg is 40 years old.
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Classify the triangle by its angle and sides
Answer:
isosceles obtuse triangleStep-by-step explanation:
isosceles obtuse triangle
An isosceles obtuse triangle is a triangle in which one of the three angles is obtuse (lies between 90 degrees and 180 degrees) and the other two acute angles are equal in measurement, and 2 equal sides.
Laura opened a deposit account. In the first month, she made an initial deposit of $2500. She plans to contribute $225 a month after her initial deposit. The account does not pay any interest.
After how many months will she have a total of $6,775?
A. 18
B. 15
C. 20
D. 21
Answer:
20 months
Turn sentences into equation:
She deposits $225 every month after initial deposit of $2500 in first month
Equation: y = 2500 + (x - 1)225
where y represents total amount, x represents months
Use this equation to find at which month her account will have $6,775
2500 + (x - 1)225 = 6,775
(x - 1)225 = 6,775 - 2,500
(x - 1)225 = 4275
x - 1 = 19
x = 20
Thus, it will take 20 months when she have a total of $6,775.
Which statement regarding the diagram is true?
m∠WXY = m∠YXZ
m∠WXY < m∠YZX
m∠WXY + m∠YXZ = 180°
m∠WXY + m∠XYZ = 180°
Which of the following determines a plane?
A. 3 non-collinear points
B. line and a point on the line
C. three collinear points
D. a straight line
Answer: A 3 non collinear pts
a plane is a flat surface like a wall
y-7 = (12-x)2 in vertex form
Answer:
y = -2x + 31
Step-by-step explanation:
Vertex form a parabola refers to a place or a point where it turns. It takes the form of;
y = mx + c
From our question, we are supposed to convert the equation y-7 = (12-x)2 into vertex form.
To begin, open the brackets on the RHS;
y - 7 = 24 - 2x.
Then move -7 to the RHS which becomes the positive.
Y = 24 - 2x + 7
= 31 - 2x
Convert it to the form y = mx + c.
y = -2x + 31
This is the vertex form, y = -2x + 31.
Note:
For a standard form/quadratic, the equation has to be in the form ax² + bx + c = y