Answer:
x = 3/2
Step-by-step explanation:
That is quite long and drawn out, so we have to get it down to a single quadratic equation, set equal to 0.
Begin by expanding the right side to get
[tex]y+3x-6=-3(x^2-4x+4)+4[/tex]
then multiplying in the -3 to get
[tex]y+3x-6=-3x^2+12x-12+4[/tex]
Now we will combine all the like terms and get everything on one side of the equals sign, and set it equal to 0:
[tex]-3x^2+9x-2=0[/tex]
In order to find the equation of the axis of symmetry we have to put it into vertex form, which is accomplished by completing the square on this quadratic.
In order to complete the square, the leading coefficient has to be a +1. Ours is a -3, so we will factor that out. First, though, now that it is set to equal 0, we will move the constant, -2, over to the other side, isolating the x terms.
[tex]-3x^2+9x=2[/tex]
Now we can factor out the -3:
[tex]-3(x^2-3x)=2[/tex]
The rules for completing the square are as follows:
Take half the linear term, square it, and add it to both sides. Our linear term is 3. Half of 3 is 3/2, and squaring that gives you 9/4. We add into the parenthesis on the left a 9/4, but don't forget about that -3 hanging around out front that refuses to be ignored. We didn't add in just a 9/4, we added in (-3)(9/4) = -27/4:
[tex]-3(x^2-3x+\frac{9}{4})=2-\frac{27}{4}[/tex]
In the process of completing the square we created a perfect square binomial on the left. Stating that binomial and simplifying the addition on the right gives us:
[tex]-3(x-\frac{3}{2})^2=-\frac{19}{4}[/tex]
We can determine the axis of symmetry at this point. Because this is a positive x-squared polynomial, the axis of symmetry is in the form of (x = ) and what it equals is the h coordinate of the vertex. Our h coordinate is 3/2; therefore, the axis of symmetry has the equation x = 3/2
Anyone know this geometry question? Will give brainiest
Twelve added to a number equals 18 find the number
Summer spent $55.60 on clothes before sales tax. If sales tax is 7.5%, what did Summer spend with tax included? *Add Explanation If U Can*
Find the distance between the two points. (2, 5) and (–1, –5)
Which points in the scatter plot are outliers?
Point A
Point F
Point H
Point K
Point M
Answer:
the points are k and f
Step-by-step explanation:
You choose 3 bananas from a group of 9. In how many ways can this be done?
Assuming the order of selection doesn't matter:
[tex] \displaystyle
\binom{9}{3}=\dfrac{9!}{3!6!}=\dfrac{7\cdot8\cdot9}{2\cdot3}=84 [/tex]
Javier is 175% heavier than his brother . if javier brother weigh 80 pounds how much do javier weigh
What is the total amount and the amount of interest earned on $6,500 at 6% for 25 years?
Answer:
Total amount earned after 25 years=$27,897.16
Step-by-step explanation:
Step 1: Express formula for calculating total amount
The total amount after 25 years can be expressed as;
A=P(1+r)^n
where;
A=total amount after 25 years
P=principal amount
r=annual interest rate
n=number of years
Step 2: Substitute values in the expression
In our case;
P=$6,500
r=6%=6/100=0.06
n=25 years
replacing in the original expression;
A=6,500(1+0.06)^25
A=27,897.16
Total amount earned after 25 years=$27,897.16
Final answer:
The total amount after 25 years will be $27,897.16, and the amount of interest earned on a $6,500 investment at a 6% annual interest rate will be $21,397.16.
Explanation:
To calculate the total amount and interest earned on $6,500 at 6% for 25 years, we'll assume that the interest is compounded annually. The formula to calculate the total amount with compound interest is:
A = P(1 + r/n)^(nt)
Where:
A is the amount of money accumulated after n years, including interest.P is the principal amount (the initial amount of money).r is the annual interest rate (in decimal).n is the number of times that interest is compounded per year.t is the time the money is invested or borrowed for, in years.Let's plug the values into the formula:
A = $6,500(1 + 0.06/1)^(1*25)
A = $6,500(1 + 0.06)^25
A = $6,500(1.06)^25
A = $6,500 * 4.29187072
A = $27,897.16
The total amount after 25 years will be $27,897.16.
To find the interest earned, subtract the principal from the total amount:
Interest = A - P
Interest = $27,897.16 - $6,500
Interest = $21,397.16
The amount of interest earned will be $21,397.16.
Classify each number as rational or irrational.
1. 0.329
2. 127.5
3. -89
4. √--- 101
Answer:
A rational number is a number with a finite number of numbers after the decimal point, or the numbers after the decimal point repeat a given patern.
An irrational number is a number with infinite numbers afther the decimal point, and those numbers have no pattern, are all "random", some examples are the roots of prime numbers, like square root of 2.
1) 0.329
It has a finite quantity of numbers after the decimal point: rational
2) 127.5
It has a finite quantity of numbers after the decimal point: rational
3) -89
It has a finite quantity of numbers after the decimal point: rational
4) [tex]\sqrt{101[/tex]
101 is a prime number, so the square root of 101 must be a irrational number.
Crater Lake Oregon is a roughly circular lake. The lake basin formed about 7000 years ago when the top of a volcano exploded in an immense explosion. Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2). Find the coordinates of the center of the lake.
Answer:
The coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].
Step-by-step explanation:
It is given that Hillman Peak, Garfield Peak, and Cloudcap are three mountain peaks on the rim of the lake. The peaks are located in a coordinate plane at H(-4,1), G(-2,-3), and C(5,-2).
If we joint these points, then we get a triangle and the center of a triangle is known as centroid.
The formula for centroid of a triangle is
[tex]Centroid=(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3})[/tex]
The centroid of the triangle H(-4,1), G(-2,-3), and C(5,-2) is
[tex]Centroid=(\frac{-4-2+5}{3},\frac{1-3-2}{3})[/tex]
[tex]Centroid=(\frac{-1}{3},\frac{-4}{3})[/tex]
Therefore the coordinates of the center of the lake are [tex](\frac{-1}{3},\frac{-4}{3})[/tex].
A sound wave travels through iron at a rate of 5120 m/s. At what rate do the sound waves travel at km/h
Answer:
Sound waves travel through iron at 11,427,840 km/h.
Step-by-step explanation:
A kilometer is equal to 0.62 miles. An hour is equal to 3600 seconds. Therefore, if a sound wave travels through iron at a rate of 5120 m/s, it would be equal to (5120 x 0.62) km/s. This is equal to 3174.4 km/s which, multiplied by 3600 (the number of seconds that make an hour), would determine the speed in which a sound wave travels through iron. So, as 3174.4 x 3600 is 11,427,840, so sound waves travel at 11,427,840 km/h.
You bought a guitar 6 years ago for $400. It's value decreases by about 13% per year. Write a formula to show the value of the guitar over those six years and determine its current value
Final answer:
The current value of the guitar after depreciating by 13% per year over 6 years is calculated using the exponential decay formula [tex]V = P(1 - r)^t[/tex], which gives us an approximate current value of $173.45
Explanation:
The given formula represents depreciation, where the value of an asset decreases over time. Applying this to the provided scenario, with an original price of $400 and a 13% annual depreciation rate, we calculate the value after 6 years. Substituting these values into the formula, we find: Value = [tex]400 \times (1 - 0.13)^6 = 400 \times 0.87^6 = \$173.45.[/tex]Therefore, after 6 years, the asset's value is approximately $173.45. Depreciation formulas like this one help assess the declining worth of assets over time due to various factors.
How do you combine like terms of #-10+7x+24-2x#?
The longer leg of a right triangle is 4 more than twice the shorter leg. The hypotenuse is 4 less than three times the short leg. What is the value of the hypotenuse?
Out of the total respondents, the percentage of respondents from the 46–55 age group who rated the film excellent is
%. Write your answer up to two decimal places.
NextReset
The percentage of respondents from the 46-55 age group who rated the film excellent is 20 %.
To determine the percentage of respondents from the 46-55 age group who rated the film excellent, you need two pieces of information: the number of respondents in that age group who rated the film excellent and the total number of respondents in that age group. The formula for calculating the percentage is:
[tex]\[ \text{Percentage} = \left( \frac{\text{Number of respondents who rated excellent}}{\text{Total number of respondents in the age group}} \right) \times 100 \][/tex]
Assuming you have these numbers, let's denote the number of respondents who rated the film excellent in the 46-55 age group as X and the total number of respondents in that age group as Y. The formula becomes:
[tex]\[ \text{Percentage} = \left( \frac{X}{Y} \right) \times 100 \][/tex]
For example, if 20 respondents in the 46-55 age group rated the film excellent out of a total of 100 respondents in that age group, the calculation would be:
[tex]\[ \text{Percentage} = \left( \frac{20}{100} \right) \times 100 = 20\% \][/tex]
Therefore, replace X and Y with the actual numbers from your data to find the specific percentage.
Which of the following is equal to 7÷9/5
A: 9×7/5 B: 7×9/5 C: 9×5/7
D: 7×5/9
Final answer:
To find the value of 7÷9/5, we multiply 7 by the reciprocal of 9/5, which is 5/9, resulting in the expression 7×5/9. The correct answer is D: 7×5/9.
Explanation:
To solve the expression 7÷9/5, we must interpret the division as multiplication by the reciprocal. In other words, dividing by a fraction is the same as multiplying by its reciprocal. Therefore, 7 divided by 9/5 is equal to 7 multiplied by the reciprocal of 9/5, which is 5/9.
So, the calculation will be 7×5/9. Here's the math:
Identify the reciprocal of 9/5, which is 5/9.
Multiply 7 by the reciprocal of 9/5: 7 × 5/9.
Calculate the product: 7 × 5 = 35, and 35 ÷ 9 does not simplify further in this context.
The correct answer from the options provided is D: 7×5/9. The other options, A: 9×7/5, B: 7×9/5, and C: 9×5/7 do not represent the correct transformation from division to multiplication by the reciprocal.
The square of a positive number decreased by twice the number is 48. Find the number
Let the required positive number is x.
The square of the number is [tex]x^2[/tex]
Now, we have been given that square of a positive number decreased by twice the number. It means we have
[tex]x^2-2x[/tex]
Now, it is equal to 48. Hence, we have
[tex]x^2-2x=48\\ x^2-2x-48=0\\ x^2-8x+6x-48=0\\ x(x-8)+6(x-8)\\ (x-8)(x+6)=0\\ x=8,-6[/tex]
The number is positive, Hence, we have [tex]x=8[/tex]
Therefore, the required number is 8
The recommended space needed for certain plant is 2 ft.² you want to plant 32 of this plants in a square garden .what is the length of each side of the garden if you follow the recommendation?
A french fry stand at the fair serves their fries in paper cones. The cones have a radius of 222 inches and a height of 666 inches. It is a challenge to fill the narrow cones with their long fries. They want to use new cones that have the same volume as their existing cones but a larger radius of 444 inches. What will the height of the new cones be?
To find the height of the new cone with a larger radius that maintains the same volume, we calculate the volume of the original cone and solve for the new height using the cone volume formula.
Explanation:The student wants to find out the new height of the cone with a larger radius while maintaining the same volume as the original cone. The volume of a cone is given by the formula V = 1/3 πr²h. To find the height of the new cone, we'll use the volume of the original cone as the constant value.
Let's first calculate the volume of the original cone:
V_original = 1/3 π(222²)(666)Now, let's set up an equation with the new radius to solve for the new height (h_new):
1/3 π(444²)(h_new) = V_originalh_new = (V_original) / (1/3 π(444²))By substituting the V_original from step 1 into step 3, we can solve for h_new. The result will be the height of the new cone with a radius of 444 inches that has the same volume as the original cone.
Determine the equation of the line given by the graph.
A) y = 2x + 4
B) y = 4x + 2
C) y = 1/2x − 2
D) y = −2x + 4
The equation of the given line is y=2x+4. Therefore, option A is the correct answer.
From the given graph coordinates on the line are (-2, 0) and (0, 4).
What is slope of a line?The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.
Slope of a line, m= 4/2 = 2
Put m=2 and (-2, 0) in y=mx+b
0=2(-2)+b
b=4
Substitute m=2 and b=4 in y=mx+b
That is, y=2x+4
The equation of the given line is y=2x+4. Therefore, option A is the correct answer.
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Find the slope of the line graphed below
A parabolic archway is 12 meters high at the vertex. at a height of 10 meters, the width of the archway is 8 meters. how wide is the archway at ground level?
We found the width at ground level to be approximately 19.6 meters.
We start by assuming the equation of the parabola to be of the form y = ax² + bx + c.
Since the vertex of the parabolic arch is at (0, 12), we know that c = 12. Next, we use the condition that at y = 10 meters, the width is 8 meters, meaning the points (4, 10) and (-4, 10) lie on the parabola. Plugging these into the equation:
For (4,10): 10 = a(4)² + b(4) + 1210 = 16a + 4b + 1216a + 4b = -2 (1)Solving for the same with (-4,10) is unnecessary as it results in the same equation. Next, we use the vertex form y = a(x - h)² + k ⇨ y = a(x - 0)² + 12 ⇨ y = ax² + 12.
From equation (1), we have 16a + 4b = -2. Since the vertex form tells us there is no 'b' term in the simplest form, 'b' must be 0. Thus, 16a = -2 ⇨ a = -1/8.
The equation of the parabola is therefore y = -(1/8)x² + 12.
To find the width at ground level (y = 0):
0 = -(1/8)x² + 12(1/8)x² = 12x² = 96 ⇨ x = ±√96 = ±4√6 ≈ ±9.8 meters.Thus, the total width of the arch at ground level is approximately 2 × 9.8 = 19.6 meters.
The graph shows a probability distribution.
What is P(1.5≤X≤4.5)?
Enter your answer, as a decimal
P(1.5≤X≤4.5) =
A swimming pool has four faucets. the first can fill the entire pool with water in two days, the second - in three days, the third - in four days, and the last one can fill the pool in 6 hours. how long will it take to fill the pool using all 4 faucets together (be exact)?
A student ate 3/20 of all candies and another 1.2 lb. Another student ate 3/5 of the candies and the remaining 0.3 lb. Altogether, what weight of candies did they eat?
A total of 243 adults and children are at a movie theater. There are 109 more adults than children in the theater. If eh represents the number of adults and b represents the number of children, which system of equations could be used to find the number of adults and the number of children in the theater?
The system of the equations representing the situation is a+b = 243 and
a-b = 109 and values of a and b are 176 and 67 respectively.
What is a system of equations?A system of equations is a collection of two or more equations with a same set of unknowns.
Given that, A total of 243 adults and children are at a movie theater. There are 109 more adults than children in the theatre. If a represents the number of adults and b represents the number of children,
Establishing the equations according to the given situation,
a+b = 243......(i)
a-b = 109.......(ii)
Adding the equations, we get,
2a = 352
a = 176
Therefore, b = 67
Hence, The system of the equations representing the situation is a+b = 243 and a-b = 109 and values of a and b are 176 and 67 respectively.
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Let v1 = (-6,4) and v^2 = (-3,6). Compute the following
V^1 * V^2
Find the perimeter of the polygon on this picture. I don’t know how to do this. Plz help
The points at which a quadratic equation intersects the x-axis are referred to as
There is a number of different things these are referred to as. The first of which is the zeroes of the equation. They are called the zeroes of the equation due to the fact that this is where the function (or y-value) is equal to 0.
It can also be referred to as the x-intercepts. This is because it is where the graph intercepts the x-axis.
The points at which a quadratic equation intersects the x-axis are referred to as x intercepts or zeros or roots of quadratic equation
Given :
The points at which a quadratic equation intersects the x-axis
The points at which the any quadratic equation crosses or touches the x axis are called as x intercepts.
At x intercepts the value of y is 0.
So , the points at which a quadratic equation intersects the x-axis is also called as zeros or roots of the quadratic equation .
The points at which a quadratic equation intersects the x-axis are referred to as x intercepts or zeros or roots of quadratic equation
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If matrix A is a 6 × 5 matrix, which order of matrix can be multiplied by matrix A?
Answer:
5 x n, where n is any natural number.
Step-by-step explanation:
If A is a 6 x 5 matrix, it means that A has 6 rows and 5 columns, this means that if a matrix B can be multiplied by A, then each one of its columns must have 5 entries. Therefore, B has to be a matrix with 5 rows and it might have any arbitrary number of columns, in other words B is a 5 x n matrix, where n must be a natural number bigger than 1.