[tex]x^2-x-42=0\\x^2+6x-7x-42=0\\x(x+6)-7(x+6)=0\\(x-7)(x+6)=0\\x=7 \vee x=-6[/tex]
Answer:
The roots are 7 and -6.
Keys:
[tex]ax^2+bx+c=0[/tex][tex]x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex][tex]-(-a)^(^n^) = a^(^n^)[/tex][tex]\sqrt[n]{a}^n = a[/tex][tex]1^a = 1[/tex]When you see a ± in a quadratic equation, you will have multiple solutions; more than one.
Step-by-step explanation:
solve the main expression
[tex]x^2-x-42=0\\x_{1,\:2}=\frac{-\left(-1\right)\pm \sqrt{\left(-1\right)^2-4\cdot \:1\cdot \left(-42\right)}}{2\cdot \:1}\\\rightarrow \sqrt{\left(-1\right)^2-4\cdot \:1\cdot \left(-42\right)}\\=\sqrt{\left(-1\right)^2+4\cdot \:1\cdot \:42}\\\rightarrow \left(-1\right)^2\\=1^2\\=1\\\rightarrow 4\cdot \:1\cdot \:42\\=168\\=\sqrt{1+168}\\=\sqrt{169}\\=\sqrt{13^2}\\=13\\x_{1,\:2}=\frac{-\left(-1\right)\pm \:13}{2\cdot \:1}[/tex]
solve for x₁
[tex]\frac{-\left(-1\right)+13}{2\cdot \:1}\\=\frac{1+13}{2\cdot \:1}\\=\frac{14}{2\cdot \:1}\\=\frac{14}{2}\\=7[/tex]
solve for x₂
[tex]\frac{-\left(-1\right)-13}{2\cdot \:1}\\=\frac{1-13}{2\cdot \:1}\\=\frac{-12}{2\cdot \:1}\\=\frac{-12}{2}\\=-\frac{12}{2}\\=-6[/tex]
Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer
The age of Jennifer is 16.
What is Jennifer's age?Let Jennifer's age be represented with x
Jonathan's age would be 2x
The sum of their ages would be 48
2x + x = 48
3x = 48
x = 48 /3 = 16
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Answer: Jennifer is 16
Step-by-step explanation: 2x+y=48, divide by 3 then multiply by 2
Given the two rectangles below. Find the area of the shaded region. 4 4 9 3
Answer:
60
Step-by-step explanation:
The shaded region's area is the difference between the area of the outer rectangle and the white rectangle.
area = LW - lw
area = (4 + 4)(9 + 3) - (4)(9)
area = (8)(12) - 36
area = 96 - 36
area = 60
What is the radius of a circle if it’s center is (1,2) and one point on the circle is (1,6)?
because the x values are identical the radius would be found by subtracting the y values.
6-2 = 4
the radius is 4
What are the fourth roots of 2√3 −2i? Enter your answer by filling in the boxes. Enter the roots in order of increasing angle measure.
By using the De Moivre's formula, the quartic roots of the complex numbers (in polar form) are z₁ = (√2, π/6), z₂ = (√2, 2π/3), z₃ = (√2, 7π/6), z₄ = (√2, 5π/3).
How to find the roots of a complex number
Complex numbers are numbers of the form a + i b, where a and b are the real and imaginary component, respectively. In other words, complex numbers are an expansion from real numbers. The n-th root of a complex number is found by using the De Moivre's formula:
[tex]\sqrt[n]{z} = \sqrt[n]{r}\cdot \left[\cos \left(\frac{\theta + 2\pi\cdot k}{n} \right) + i\,\sin \left(\frac{\theta + 2\pi\cdot k}{n}\right)\right][/tex], for i = {0, 1, 2, ..., n - 1}.
Where:
r - Norm of the complex number.θ - Direction of the complex number, in radians.The norm of the complex number is found by Pythagorean theorem:
[tex]r = \sqrt{(2\sqrt{3})^{2}+(-2)^{2}}[/tex]
r = 4
And the direction is determined below:
[tex]\theta = \tan^{-1} \left(\frac{2\sqrt{3}}{-2} \right)[/tex]
θ = 2π/3 rad
Then, the quartic roots of the complex numbers (in polar form) are z₁ = (√2, π/6), z₂ = (√2, 2π/3), z₃ = (√2, 7π/6), z₄ = (√2, 5π/3).
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A box contains 50 batteries, of which 10 are dead and 5 are weak. Suppose you select batteries at random from the box and set them aside for recycling if they are dead or weak. If the first battery you select is dead and the second one is weak, what is the probability that the next battery you select will be weak?
what type of parent function is f(x) = |x|
Answer: Absolute value function
The type of parent function is f(x)= |x| is Absolute valued function
What is Absolute valued function?
An absolute value function is a function that contains an algebraic expression within absolute value symbols.
Given function:
f(x)= |x|
As, we know the modulus generally give the absolute value of any function and also the modulus always give the positive value.
The above property perfectly resemble the absolute valued function.
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Which lists all of the x-intercepts of the graphed
function?
O (0, 6)
O (1, 0) and (2, 0)
O (1, 0), (2, 0), and (-3, 0)
O (1, 0), (2, 0), (-3, 0), and (0, 6)
Answer:
(1, 0), (2, 0), and (-3, 0)
Step-by-step explanation:
The x intercepts are where it crosses the x axis. It crosses at (-3,0), (1,0) and (2,0)
You have been given data about gas prices in 15 states. Comparing the two data sets below, what does the larger standard deviation, range, and variance say about gas prices in 2022 compared to gas prices in 2018?
Gas Prices in 2018 Gas Prices in 2022
Max: $2.42
Min: $2.03
Mean: $2.098
Median: $2.06
Mode: $2.04
Variance: 0.0121
Standard Deviation: $0.11
Max: $5.87
Min: $4.48
Mean: $4.83
Median: $4.72
Mode: $4.72
Variance: 0.0142
Standard Deviation: $0.38
Comparing the two data sets shown below
What is standard deviation and variance?Variance is the average squared deviations from the mean, while standard deviation is the square root of this number.
Both the mean and the median can be used to describe where the “center” of a dataset is located.
So, the mean of data in 2022 is more than mean of data in 2018.
It’s best to use the mean when the distribution of the data values is symmetrical and there are no clear outliers.
As the variance of 2022 more than in variance in 2018.
A high variance indicates that the data points are very spread out from the mean, and from one another.
As the standard deviation of 2022 more than in standard deviation in 2018.
The high standard deviation indicates data are more spread out.
The bigger the range, the more spread out the data.
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Consider triangle ABC. What is b?
Answer:
A triangle is equal to 540° so you do 112 add 37 which equals 149° then take it away from 540 which gives you 391° as your answer
Step-by-step explanation:
Your Welcome
A pre-school has 12 four-year old children and 11 three-year old children. What is the ratio of four-year old
children to three-year old children?
Answer:
12:11
Step-by-step explanation
Answer:
I think it would be 12:11
Step-by-step explanation:
This is because you can't down size the 11
Select the correct answer.
One leg of a right triangle is 10 units, and its hypotenuse is 12 units. What is the length of the other leg?
A.
about 7 units
B.
about 15 units
C.
about 25 units
D.
about 122 units
E.
about 224 units
Anyone mind helping me solve this inequality? Step by step
By decomposing the figure in simpler shapes, we will see that the total area is:
a = 180 cm²
How to find the area of the composite figure?
Remember that the area of a rectangle of width W and length L is:
A = L*W
And the area of a triangle with base B and height H is:
A = B*H/2.
Then, the upper part can be seen as a rectangle of length of 6cm and width of 6 cm, with two triangles on the sides, such that each triangle has a base of 3cm and a height of 6cm.
So the area of that part is:
A = 6cm*6cm + 2*(3cm*6cm/2) = 54cm²
Now, the bottom triangle has a base of 12 cm, and a height of:
15cm - 6cm = 9cm
Then its area is:
A' = 12cm*9cm/2 = 54cm²
This means that the total area of the figure is:
total area = 54cm² + 54cm² = 108cm²
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Please help asap 25 points
Answer:
[tex]y = - 2 |x - 2| + 4[/tex]
Step-by-step explanation:
as the function moved to the right 2 steps, and up 4 steps
The weight, w, that a horizontal beam can support varies inversely as the
length, l, of the beam. a beam measuring 5 meters in length can support
430 kilograms. how much weight can a beam measuring 2 meters
support?
A beam measuring 2 meters can support 172 kilograms
How to determine the weight?The variation is a direct variation.
So, we have:
w = kl
Where k is the constant of variation.
When l = 5, w = 430.
This means that:
430 = 5k
Divide both sides by 5
k = 86
So, we have:
w = 86l
When l = 2, we have:
w = 86 * 2
Evaluate
w = 172
Hence, a beam measuring 2 meters can support 172 kilograms
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Given quadrilateral A(2,-1) B(1,3) C(6,5) D(7,1), you want to prove that it is a parallelogram by showing opposite sides are parallel. What formula would you use?
Answer: Slope formula
Step-by-step explanation:
You could show the slopes of the opposite sides are equal, and thus they are parallel.
Which expression is equivalent to
4^sqrt 6 / 3^sqrt2
Answer: Choice C
[tex]\frac{\sqrt[12]{55,296}}{2}[/tex]
==========================================================
Explanation:
Recall the process to rationalize the denominator is to multiply top and bottom by the denominator expression. This only works if we have a square root in the denominator.
For cube roots, we'll need to multiply two copies of the denominator to top and bottom. This will get us three copies of [tex]\sqrt[3]{2}[/tex] so that the cube root goes away.
Check out the steps below to see what I mean:
[tex]\frac{\sqrt[4]{6}}{\sqrt[3]{2}}\\\\\\\frac{\sqrt[4]{6}\sqrt[3]{2}\sqrt[3]{2}}{\sqrt[3]{2}\sqrt[3]{2}\sqrt[3]{2}}\\\\\\\frac{6^{1/4}*2^{1/3}*2^{1/3}}{\left(\sqrt[3]{2}\right)^3}\\\\\\\frac{6^{3/12}*2^{4/12}*2^{4/12}}{2}\\\\\\\frac{\left(6^3*2^4*2^4\right)^{1/12}}{2}\\\\\\\frac{\left(55,296\right)^{1/12}}{2}\\\\\\\frac{\sqrt[12]{55,296}}{2}\\\\\\[/tex]
In short,
[tex]\frac{\sqrt[4]{6}}{\sqrt[3]{2}}=\frac{\sqrt[12]{55,296}}{2}[/tex]
We have the 12th root of 55,296 all over top the integer 2. The "2" is not part of the 12th root. This shows why choice C is the final answer.
Answer:
C
Step-by-step explanation:
Choose the graph below that matches the equation: =2/3 + 5
It is TRUE, so shade on the SAME side as (0,0)
Each pepperoni slice is 78 of an inch long and costs $0.03. Exactly enough pepperoni slices are laid end to end to have a total length of 56 inches. What is the total cost of these pepperoni slices?
The total cost of these pepperoni slices is $1.92
What is the total cost of these pepperoni slices?
Each slice measures 7/8 of an inch. Here we have N slices, such that the measure of the N slices is 56 inches, then we have:
N*(7/8 in) = 56 in
Now we can find the value of N, the number of slices used:
N = (8/7)*56 = 64
And each slice costs $0.03, then the cost of the 64 slices is:
C = $0.03*64 = $1.92
The total cost of these pepperoni slices is $1.92
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Complete the equivalent equation for –7x – 60 = x2 + 10x.
(x +
)(x +
) = 0
By algebraic handling and factoring, we conclude that the equivalent equation for the expression - 7 · x - 60 = x² + 10 · x is (x + 5) · (x + 12) = 0.
How to determine the factor form of the given equation
In this question we have a quadratic function which must be factored by algebraic handling:
- 7 · x - 60 = x² + 10 · x
x² + 10 · x + 7 · x + 60 = 0 Compatibility with addition/Modulative property
x² + 17 · x + 60 = 0 Definition of addition
(x + 5) · (x + 12) = 0 x² + (- a - b) · x + a · b = 0/Result
By algebraic handling and factoring, we conclude that the equivalent equation for the expression - 7 · x - 60 = x² + 10 · x is (x + 5) · (x + 12) = 0.
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The amount of parts, p, that a shop produced in d days is shown in the table below.
d
1 2 3 4 5
p(d) 66 132 240 326 455
Select the average rate of change for the number of parts that the company produced from day 1 to day 3.
87
91
66
146
Using it's concept, it is found that the average rate of change for the number of parts that the company produced from day 1 to day 3 is of 87.
What is the average rate of change?It is given by the change in the output divided by the change in input.
In this problem, in 2 days, from day 1 to day 3, the number of parts increased from 66 to 240, hence the rate of change is given by:
r = (240 - 66)/2 = 87.
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Please Help I Don't Understand!
Answer:
a reduction
Step-by-step explanation:
this is because its growing smaller
The radius of a wheel on a toy truck is 4 inches. To the nearest foot, how far does the wheel
travel when it makes 7 revolutions?
find the value of a 3a+2b=8 and b=-2
Answer:
The value of a in the given equation is '4'
Step-by-step explanation:
The given equation is: 3a + 2b = 8 and b = -2
Plug the value of b into the formula:
3a + 2(-2) = 8
3a - 4 = 8
We want 3a by itself so we will add +4 to both sides:
3a - 4 + (4) = 8 + (4)
3a = 12
Divide both sides by 3 so a is by itself:
3a/3 = 12/3
1a = 4
a = 4
5x-4y=10
slope intercept form
Answer:
5x-4y=10
solution
first find x intercept is the graph cross x-axis.to find the x intercept,we set y¹=0 and the Solve x.
5x-4y=10
5x-4(0)=0
x¹ =2 y¹=0
y intercept
5x-4y=10
5(0)-4y=10
y²=2.5 x²=0
Find the value of z if the volume of the cylinder is 1696.5m^3 use 3.14 for pi and round your answer to the nearest whole number.
Answer:
6 m
Step-by-step explanation:
The missing dimension can be found using the volume formula with the known values filled in. This gives an equation that can be solved for z.
The volume of a cylinder is given by the formula ...
V = πr²h
__
Using the given values, we have ...
1696.5 m³ = 3.14(z²)(15 m)
z² = (1696.5 m³)/(47.1 m) ≈ 36.0191 m² . . . . divide by 47.1 m
Taking the square root, we find z to be ...
z ≈ √(36.0191 m²) ≈ 6.00 m
The value of z is about 6 m.
The histogram shows the results of a survey that asked people how many hours of sleep they get each night. Which
statements are true?
The true statements are
The mean is near the median.
The mean is the best measure of center.
The five-number summary is the best measure of variation
What is a histogram?
A histogram is used to represent data graphically. The histogram is made up of rectangles whose area is equal to the frequency of the data and whose width is equal to the class interval.
If the mean is greater than the median, the histogram would be skewed to the right. If the mean is less than the median, the histogram would be skewed to the left.
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Can anyone help me i really need it rn
Answer:
11. g
12. f
13. e (add all the frequencies)
14. d
15. c (eg. 17.5 -8.5 = 9)
16. b
17. a
Let me know if you have any questions.
Hope this helps!
Find the areas of the trapezoids.
Answer:
7.5 sq. un
Step-by-step explanation:
Hello!
The height is 3 units, the base is 4 units, and the other base is 1 unit.
Using the trapezoid area formula, [tex]A = \frac12(b_1+b_2)h[/tex], we can plug the values in and solve.
Solve:
[tex]A = \frac12(b_1+b_2)h[/tex][tex]A = \frac12(1 + 4)3[/tex][tex]A = \frac12(5)(3)[/tex][tex]A = 2.5(3)[/tex][tex]A = 7.5[/tex]The area is 7.5 sq. un
I need help on this 1 question. Thank you.
Answer: 5/6, 3/4, 4/12
Step-by-step explanation:
Convert them all into a common denominator, 12.
5/6=10/12, 3/4=9/12, and 4/12 stays the same. 10>9, and 9>4
Given that XW bisects AXZ, what theorem or theorems could be used to justify that W is equidistant from the two outside rays?
Answer:
HA congruence postulate
Step-by-step explanation:
In order to show W is equidistant from A and Z, we must demonstrate that WA ≅ WZ. The HA congruence postulate can be used for that purpose.
__
proofWX bisects AXZ, so ∠WXA≅∠WXZ by the definition of an angle bisector. By the reflexive property of congruence, WX≅WX. The angles at A and Z are given as right angles.
So, we have triangles WAX and WZX that are right triangles with congruent hypotenuse and corresponding angle. These are the preconditions for the application of the HA congruence postulate. That postulate says the triangles WAX and WZX are congruent, so ...
WA ≅ WZ by CPCTC. Hence W is equidistant from the outside rays.
__
Additional comment
The definition of distance from a point to a line is the perpendicular distance from the point to the line. In this geometry, the perpendicular distances from W to the outside rays are the lengths of segments WA and WZ. That is why we need to show those lengths are the same.