The x-intercepts of the parabola are [tex]\(x = -3 + \sqrt{\frac{14}{3}}\) and \(x = -3 - \sqrt{\frac{14}{3}}\).[/tex]
To find the x-intercepts of the parabola, we need to set y=0 in the equation of the parabola and solve for x.
Given that the vertex of the parabola is [tex]\((-3, -14)\),[/tex] the equation of the parabola can be expressed in the form [tex]\(y = a(x - h)^2 + k\)[/tex], where [tex]\((h, k)\)[/tex]represents the vertex and a is the coefficient determining the direction and width of the parabola.
Using the vertex form, we have:
[tex]\[ y = a(x + 3)^2 - 14 \][/tex]
We know that the y-intercept is (0,13) so when x=0, y=13:
[tex]\[ 13 = a(0 + 3)^2 - 14 \]\[ 13 = a(9) - 14 \]\[ 13 = 9a - 14 \]\[ 9a = 13 + 14 \][/tex]
[tex]\[ 9a = 27 \]\[ a = \frac{27}{9} \]\[ a = 3 \][/tex]
So, the equation of the parabola is:
[tex]\[ y = 3(x + 3)^2 - 14 \][/tex]
Now, to find the x-intercepts, we set y=0:
[tex]\[ 0 = 3(x + 3)^2 - 14 \]\[ 3(x + 3)^2 = 14 \]\[ (x + 3)^2 = \frac{14}{3} \][/tex]
Now, we take the square root of both sides:
[tex]\[ x + 3 = \pm \sqrt{\frac{14}{3}} \][/tex]
Rita has taken out a loan for $2000 to help pay for a car. The 2-year loan has 12% simple annual interest. What is the total amount of money that she will have paid back at the end of two years
Answer:
it should be $2480Step-by-step explanation:
Answer:
The Answer IS: 2480 I just did the question
Step-by-step explanation:
Find the inverse of the given function. f(x)= -1/2SQR x+3, x greater than or equal to -3
Answer:
So, the inverse of function
[tex]f(x) = \frac{-1}{2} \sqrt{x+3}[/tex] is [tex]f^{-1}(x)= 4x^2-3[/tex]
Step-by-step explanation:
We need to find the inverse of the given function
[tex]f(x) = \frac{-1}{2} \sqrt{x+3}[/tex]
To find the inverse we replace f(x) with y
[tex]y = \frac{-1}{2} \sqrt{x+3}[/tex]
Now, replacing x with y and y with x
[tex]x = \frac{-1}{2} \sqrt{y+3}[/tex]
Now, we will find the value of y in the above equation
Multiplying both sides by -2
[tex]-2x = \sqrt{y+3}[/tex]
Taking square on both sides
[tex](-2x)^2 = (\sqrt{y+3})^2[/tex]
[tex]4x^2 = y+3[/tex]
Finding value of y
[tex]y = 4x^2-3[/tex]
Replacing y with f⁻¹(x)
[tex]f⁻¹(x)= 4x^2-3[/tex]
So, the inverse of function
[tex]f(x) = \frac{-1}{2} \sqrt{x+3}[/tex] is [tex]f^{-1}(x)= 4x^2-3[/tex]
ANSWER
[tex]f^{ - 1} (x) =4 {x}^{2}- 3[/tex]
EXPLANATION
A function will have an inverse if and only if it is a one-to-one function.
The given function is
[tex]f(x) = - \frac{1}{2} \sqrt{x + 3} \: \: where \: \: x \geqslant - 3 [/tex]
To find the inverse of this function, we let
[tex]y=- \frac{1}{2} \sqrt{x + 3}[/tex]
Next, we interchange x and y to get,
[tex]x=- \frac{1}{2} \sqrt{y+ 3}[/tex]
We now solve for y.
We must clear the fraction by multiplying through with -2 to get;
[tex] - 2x = \sqrt{y + 3} [/tex]
Square both sides of the equation to get:
[tex](- 2x)^{2} = (\sqrt{y+ 3}) ^{2} [/tex]
[tex]4x^{2} = y + 3[/tex]
Add -3 to both sides
[tex]4 {x}^{2} - 3 = y[/tex]
Or
[tex]y = 4 {x}^{2}- 3[/tex]
This implies that,
[tex]f^{ - 1} (x) =4 {x}^{2}- 3[/tex]
This is valid if and only if
[tex]x \geqslant - 3[/tex]
The slope of the line containing the points (6, 4) and (-5, 3) is:
A. 1/11
B. 1
C. -1
[tex]\bf (\stackrel{x_1}{6}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{-5}~,~\stackrel{y_2}{3}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{3-4}{-5-6}\implies \cfrac{-1}{-11}\implies \cfrac{1}{11}[/tex]
Answer:
1/11
Step-by-step explanation:
what is 3log(2)3-log(2)(x+4) written as a single logarithm
Answer:
[tex]\large\boxed{\log_2\dfrac{x^3}{\left(\dfrac{3}{x+4}\right)}}[/tex]
Step-by-step explanation:
[tex]\text{Use}\\\\\log_ab^n=n\log_ab\\\\\log_a\left(\dfrac{c}{d}\right)=\log_ac-\log_ad\\\\==============================[/tex]
[tex]\text{We have:}\\\\3\log_2x-(\log_23-\log_2(x+4))\\\\=\log_2x^3-\log_2\dfrac{3}{x+4}\\\\=\log_2\dfrac{x^3}{\frac{3}{x+4}}[/tex]
The expression 3log(2)3 - log(2)(x+4) can be simplified by using the logarithmic laws, resulting in the final single logarithm of log2(39 / (x+4)).
Explanation:The original expression, 3log(2)3 - log(2)(x+4), can be simplified using a few laws of logarithms. Firstly, using the rule that the logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, we can simplify. We find that the expression becomes log(2)33 - log(2)(x+4). Secondly, by applying the rule that the logarithm of the number resulting from the division of two numbers is the difference between the logarithms of the two numbers, we can further simplify. The final expression is now log2(39 / (x+4)). This is the expression written as a single logarithm.
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if P is 30 units and I is 10 units, w is
Note that perimeter of rectangle [tex]P=2l+2w[/tex]
So we have to solve for width or w.
[tex]P=2l+2w[/tex]
[tex]30=2\times10+2w[/tex]
[tex]w=5[/tex]
Width of a rectangle is 5 units.
Hope this helps.
r3t40
what is the solution for 3/2 = 3/2x - 6/5x
Answer:
x=5
Step-by-step explanation:
3/2 = 3/2x - 6/5x
Get a common denominator of 10 for the fractions with x
3/2 = 3/2 *5/5 x - 6/5 *2/2 x
3/2 = 15/10x - 12/10x
3/2 = 3/10 x
Multiply each side by 10/3 to isolate x
10/3 * 3/2 = 10/3 * 3/10 x
10/2 = x
5 =x
Write a phrase as an algebraic expression
4 times the sum of a number and 20
A 20 / y C 4(y + 20)
B 20 + y D 4y - 20y
Answer:
I believe it's C.
Step-by-step explanation:
Hope my answer has helped you!
assume a*b means a+b-1. what is 5*3
Answer:
7
Step-by-step explanation:
Replace a with 5 and b with 3.
5+3-1=7
Answer:
7.
Step-by-step explanation:
Substitute for a and b in the given identity:
5 * 3 = 5 + 3 - 1
= 8 - 1
= 7.
The common ratio is 2
40, 20, 10, 5,...
Answer:
2.5
Step-by-step explanation:
because half of 5 is 2.5
what is .84 /.02?
i really need help with this question so please help me!
Answer:
42
Step-by-step explanation:
This means 0.84 divided by 0.02
which equals to 42
the points mean that there is a zero
if a number has been written in that format of "point" then "number" without any number before the point, it means there is a zero there
if you need anything else clarified please do so in the comment section i would like to help more
Hope this helps and if it does mark as branliest answer thx
The graph of y=x^3 is transformed as shown below?
Solution:
A
Step-by-step explanation:
It's y =-2x³
Hope it helps you..☺
The graph of y = x³ is transformed as y = -6x³.
What is a function?It's a unique type of connection with a predetermined domain and range, and every value in the domain exists associated with precisely one value in the range, according to the function.
The given equation exists, y = x³
The graph of y = x³ describes the transformed function.
Therefore, the correct answer is option B. y = -6x³.
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Solve 9x + 4 = 11 for x using the change of base formula log base b of y equals log y over log b
Final answer:
To solve the equation 9x + 4 = 11 for x using the change of base formula log base b of y equals log y over log b, isolate x on one side of the equation, subtract 4 from both sides, then divide both sides by 9.
Explanation:
To solve the equation 9x + 4 = 11 for x using the change of base formula log base b of y equals log y over log b, we need to isolate x on one side of the equation. Here's how to do it:
Subtract 4 from both sides of the equation: 9x = 11 - 4 = 7.Divide both sides of the equation by 9: x = 7/9.Therefore, the solution to the equation is x = 7/9.
Adam solved this equation and identified the number of solutions.
24x – 22 = 4(6x – 1)
24x – 22 = 24x – 4
24x = 24x + 18
0 = 18
The equation has infinitely many solutions
what is his mistake
Answer:
The mistake is he said"The equation has infinitely many solutions."
But based on what he was solving for, this equation does not have any solutions, so that's his mistake.
The other answer is correct, but the answer is D if you just want the answer.
Factor the following expression completely. 16x^5-x^3
A.
B.
C.
D.
Answer:
x^3(2x+1)(2x-1)
Step-by-step explanation:
first take common then use formula of a^2-b^2
Answer:
The factor of the provided expression are: [tex]x^3(4x-1)(4x+1)[/tex]
Step-by-step explanation:
Consider the provided expression.
[tex]16x^5-x^3[/tex]
Here the Greatest common factor in the above expression is x³.
The above expression can be written as:
[tex]x^3(16x^2-1)[/tex]
[tex]x^3((4x)^2-1^2)[/tex]
Now use the difference of the square property: [tex]a^2-b^2=(a+b)(a-b)[/tex]
By using the above property we can rewrite the provided expression as shown:
[tex]x^3(4x-1)(4x+1)[/tex]
Hence, the factor of the provided expression are: [tex]x^3(4x-1)(4x+1)[/tex]
what's 2,000 times 4,000???
Answer:
8,000,000
Step-by-step explanation:
all you have to do is multiply the 4*2 then add the missing 0's for example so its 8 then add the six zeros behind the 8
The value of the multiplication is 8, 000 , 000
How to determine the valueFirst, we need to know that multiplication is an arithmetic operation.
Also, PEDMAS is the mathematical acronym used to represent the different arithmetic operations.
They are given as;
P represents parenthesesE is for exponentsM is for multiplicationD is for divisionS is for subtractionFrom the information given, we have that;
2,000 times 4,000
This is represented as;
2000 ×4000
Multiply the values
8, 000 , 000
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How long is AC
Angle B =60°
And BC = 3√ 3
Answer:
AB or x =6sqrt(3) (which is what the pic ask for)
but you ask for AC which is 9
Step-by-step explanation:
The hypotenuse is twice the length of the short leg.
The short leg is 3sqrt(3) (because I see it is opposite the smallest angle which is 30 in this case).
So the hypotenuse is twice 3sqrt(3) which is 6sqrt(3)
You are also asking for AC though... That is the short leg times sqrt(3) so that measurement is 3sqrt(3)*sqrt(3)=3(sqrt(3*3))=3(3)=9
How is the product of 2 and –5 shown using integer tiles?
Answer:
Mykel you should have 2 rows of -5
Step-by-step explanation:
so the first row should have 5 red blocks and the second row should also have 5 red blocks. The reason it should be red is because red stands for negative.
The graph above shows Carmel's distance from home over a one-hour period, during which time he first went to the library, then went to the grocery store, and then returned home. Which of the following statements could be true?
A) The grocery store is about 5 miles from Carmel's house.
B) Carmel traveled a total of 7 miles from the time he left home until he returned.
C) The grocery store is 7 miles farther from Carmel's house than the library is.
D) Carmel spent 10 minutes at the library and 15 minutes at the grocery store.
Answer:
D
Step-by-step explanation:
From the graph we can see:
The library is 5 miles from home (upward sloping line)He spends 10 minutes at library (sraight line for 10 minutes)The grocery store is 2 miles from library , hence 7 miles from homeHe spends 15 minutes at grocery store (straight line for 15 minutes)Returns back home (7 miles)Given the information extract, we can rule out A, B, and C immediately. Answer choice D is right.
You need a 15% alcohol solution. On hand, you have a 210 mL of a 35% alcohol mixture. How much pure water will you need to add to obtain the desired solution?
Answer:
280 mL
Step-by-step explanation:
If x is the amount of 35% solution and y is the amount of pure water, then:
0.15 = (0.35x + 0y) / (x + y)
0.15 = (0.35x) / (x + y)
Given that x = 210 mL:
0.15 = (0.35×210) / (210 + y)
210 + y = 490
y = 280
One x-intercept for a parabola is at the point
(0.2,0). Use the quadratic formula to find the
other x-intercept for the parabola defined by
this equation:
y = 5x² + 4x - 1
Separate the values with a comma. Round, if
necessary, to the nearest hundredth.
Answer:
x=-1 or (-1,0)
Step-by-step explanation:
(-4 +- sqrt(4^2-4(5)(-1)))/2*5
(-4 +- sqrt(16+20))/10
(-4 +- sqrt(36))/10
(-4 +- 6)/10
x= (-4+6)/10 = 2/10 = 1/5 = 0.2
x= (-4-6)/10 = -10/10 = -1
x-intercepts: (0.2,0), (-1,0)
If f(x) = -x + 8 and g(x) = x4, what is (gºf)(2)?
Enter the correct answer.
Answer:
[tex]\large\boxed{(g\circ f)(2)=1296}[/tex]
Step-by-step explanation:
[tex]f(x)=-x+8,\ g(x)=x^4\\\\(g\circ f)(x)=g\bigg(f(x)\bigg)\\\\(g\circ f)(x)=\bigg(-x+8\bigg)^4\\\\(g\circ f)(2)\to\text{put x - 0 to the equation of the function:}\\\\(g\circ f)(2)=(-2+8)^4=(6)^4=1296[/tex]
Using the graph below, what is the best estimate of the hourly rate a person
earns for 5 years of experience if the equation for the line of best fit is
y= 1.3x +5.5?
Hourly rate ($)
Years of experience
A. $13.00/hr
B. $12.00/hr
C. $10.50/hr
D. $12.50/hr
Answer:
B
Step-by-step explanation:
x is the number of years of experience.
In the graph, the point (y) is not shown for 5 years of experience but we can easly figure this out by plugging in 5 into x of the equation of best fit given.
Hence
[tex]y= 1.3x +5.5\\y= 1.3(5) +5.5\\y=12[/tex]
So, the best estimate is $12 per hour, the correct answer is B
Answer:
its b
hope its right
Simplify the following expression:
18g+7-12g-3
Answer:
[tex]18g + 7 - 12g - 3 = 6g + 4[/tex]
Step-by-step explanation:
[tex]18g+7-12g-3\\=18g-12g+7-3\\=6g+7-3\\=6g+4[/tex]
A. 530 ft^2
B. 500 ft^2
C. 470 ft^2
D. 450 ft^2
Answer: OPTION A.
Step-by-step explanation:
Find the length scale factor by dividing the known length of the larger triangle by the known length of the smaller triangle:
[tex]lenght\ scale\ factor=\frac{35}{25}=\frac{7}{5}[/tex]
Then, the area scale factor is:
[tex]area\ scale\ factor=(\frac{7}{5})^2=\frac{49}{25}[/tex]
To find the area of the the larger triangle, multiply the area of the smaller triangle by the area scale factor. Then:
[tex]A_{(larger)}=(270ft^2)(\frac{49}{25})=529.2ft^2[/tex]
So, the option that shows an approximation of the area of the larger triangle is the option A:
[tex]A_{(larger)}[/tex]≈[tex]530ft^2[/tex]
A force of 60 N is used to stretch two springs that are initially the same length. Spring A has a spring constant of 4 N/m, and spring B has a spring constant of 5 N/m.
How do the lengths of the springs compare?
A:Spring B is 1 m longer than spring A because 5 – 4 = 1.
B:Spring A is the same length as spring B because 60 – 60 = 0.
C:Spring B is 60 m longer than spring A because 300 – 240 = 60.
D:Spring A is 3 m longer than spring B because 15 – 12 = 3.
Answer:
D:Spring A is 3 m longer than spring B because 15 – 12 = 3.
Step-by-step explanation:
In this question, you should remember the Hooke's Law in physics.
The Hooke's Law simply explains that the extension that occurs on a spring is directly proportional to the load applied on it.
The mathematical expression for this law is
[tex]F=-kx[/tex]
where;
F= force applied on the spring
x = the extension on the spring
k= the spring constant which varies in spring.
The question will need you to calculate the extension on the springs A and B then compare the values obtained.
In spring A
Force, F=60N and spring constant ,k=4 N/m
To find the extension x apply the expression;
[tex]F=-kx\\\\60=-4*x\\\\60=-4x\\\\\frac{60}{-4} =\frac{-4x}{-4} \\\\\\-15=x[/tex]
Here the spring extension is 15 m
In spring B
Force, F=60N and spring constant , k=5N/m
To find the extension x apply the same expression
[tex]F=-kx\\\\60=-5*x\\\\60=-5x\\\\\\\frac{60}{-5} =\frac{-5x}{-5} \\\\\\-12=x[/tex]
Here the extension on the spring is 12 m
Compare
The extension on spring A is 3 m longer than that in spring B because when you subtract the value of spring B from that in spring A you get 3m
[tex]=15m-12m=3m[/tex]
Answer:
D)Spring A is 3 m longer than spring B because 15 – 12 = 3.
trust
what are the length and width of a rectangular traffic sign if the length exceeds the width by 24 inches and the perimeter by 128 inches
Answer:
length = 44 in, width = 20 inStep-by-step explanation:
[tex]\text{The formula of a perimeter of a rectangle:}\\\\P=2(l+w)\\\\l-length\\w-width\\\\\text{We have}\ l=(w+24)\ in,\ P=128\ in\\\\\text{Substitute:}\\\\128=2(w+24+w)\qquad\text{divide both sides by 2}\\\\64=2w+24\qquad\text{subtract 24 from both sides}\\\\40=2w\qquad\text{divide both sides by 2}\\\\20=w\to w=20\ in\\\\l=20+24=44\ in[/tex]
If the following ordered pairs are equal find x and y
a) (3x+4y, 10) and (20, 40-3y)
Step one: 3x+4y=20; 10=40-3y
Step two: 3y=40-10, so y=(40-10)/3=10
Step three: 3x+4*10=20
Step four: 3x=20-40=-20, so x=-20/3=-6.66666...~ -6.67
10x®y 12
Which expression is equivalent to 7-2.. 6? Assume X+0.y* 0.
For this case we must find an expression equivalent to:
[tex]\frac {10x ^ 6y ^ {12}} {- 5x ^ {- 2} y^ {- 6}}[/tex]
We have to:
[tex]\frac {10} {- 5} = - 2[/tex]
Now, by definition of division of powers of equal base, we put the same base and subtract the exponents. Rewriting the expression we have:
[tex]-2x ^ {6 - (- 2)} y^ {12 - (- 6)} =\\-2x ^ {6 + 2} y ^ {12 + 6} =\\-2x ^ 8y ^ {18}[/tex]
Answer:
Option B
Consider the binomial 4x3 – 32. Is there a GCF > 1 for the two terms? The completely factored form of this polynomial is (x2 + 2x + 4).
Answer:
Yes there is GCF > 1 ⇒ (GCF = 4)
The completely factored form is 4(x - 2)(x² + 2x + 4)
Step-by-step explanation:
* Lets find the greatest common factor of the two terms
- The binomial is 4x³ - 32
- The terms of the binomial are 4x³ and 32
- The greatest common factor of 4 and 32 is 4 because both of them
can divided by 4
∵ 4x³ ÷ 4 = x³
∵ 32 ÷ 4 = 8
∴ The greatest common factor GCF is 4
∴ 4x³ - 32 = 4(x³ - 8)
* Yes there is GCF > 1
# x³ - 8 is the difference of two cubs, it can factorize it into two
brackets
- The first bracket has cube root of x³ and cube root of 8
- The second bracket comes from the first bracket it has three terms
# The 1st term is square the 1st term in the first bracket
# The 2nd term is the product of the 1st term and the 2nd term of the
1st bracket with opposite sign of the 2nd term in the 1st bracket
# The 3rd term is the square of the 2nd term in the 1st bracket
* Lets do these steps with x³ - 8
∵ The first bracket = (∛x³ - ∛8)
∵ ∛x³ = x and ∛8 = 2
∴ The first bracket = (∛x³ - ∛8) = (x - 2)
- Lets make the 2nd bracket from the 1st bracket
∴ The second bracket = (x² + (x)(2) + 2²)
∴ The second bracket = (x² + 2x + 4)
∴ The factorization of x³ - 8 = (x - 2)(x² + 2x + 4)
* The completely factored form is 4(x - 2)(x² + 2x + 4)
Answer:
....Yes it is 4 ....4(x-2)
Step-by-step explanation:
the screenshot shows proof :))
Use long division or synthetic division to find the quotient of 2x^3+x^2+1 / x+1
Answer:
The quotient is 2x^2 - x + 1.
Step-by-step explanation:
If x + 1 is the divisor in long division, then -1 is the divisor in synthetic division:
-1 / 2 1 0 1
-2 1 -1
----------------------
2 -1 1 0
The quotient is 2x^2 - x + 1. The coefficients were obtained through synthetic division.