Using complex numbers, you can have negative squares, since [tex] i^2=-1 [/tex]
So, you can find the solutions to
[tex] x^2+20=0 \iff x^2 = -20 \iff x=\pm\sqrt{-20}=\pm i \sqrt{20}=\pm 4i\sqrt{5} [/tex]
So, using the factorization [tex] (x-x_1)(x-x_2) [/tex] where [tex] x_{1,2} [/tex] are the roots of the quadratic equation, you have
[tex] x^2+20 = (x-4i\sqrt{5})(x+4i\sqrt{5})[/tex]
This expression can't be further factorized, because all terms have degree 1.
To factor the expression x^2 + 20 over the complex numbers, we can use the quadratic formula or complete the square.
Explanation:To factor the expression x^2 + 20 over the complex numbers, we can use the quadratic formula or complete the square.
Using the quadratic formula, we start by finding the values of x that satisfy the equation x^2 + 20 = 0.
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a).
In this case, a = 1, b = 0, and c = 20. Substituting these values into the quadratic formula, we get: x = ± √-20.
Since we are factoring over the complex numbers, we can express the solution as: x = ± √20i.
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If f(x) = 3x and g(x) = 1/3x which expression could be used to verify that g(x) is the inverse of f(x)?
A. 3x(x/3)
B. (1/3x)(3x)
C.1/3(3x)
D. 1/3(1/3x)
If we want to verify that g ( x ) is the inverse of f ( x ) we have to show that:
( f ° g ) = ( g ° f )
f ( g ( x ) )= 3 · ( 1/3 x ) = x
g ( f ( x ) ) = 1/3 · ( 3 x ) = x
Answer:
C ) 1/3 ( 3 x )
Answer:
C. 1/3(3x)Step-by-step explanation:
To verify if two functions are inverse, we have to find the composition of such functions, that is:
[tex]f(g(x))[/tex]; which results must be the variable [tex]x[/tex] to be inverse functions.
We know that: [tex]f(x)=3x;g(x)=\frac{1}{3}x[/tex].
Replacing on the composition:
[tex]f(g(x))=3(\frac{1}{3}x)=x[/tex]
Therefore, the expression that has to be used to verify that these functions are inverse is C.
Find the value of x (3x+31)° (2x-6)°
Angle (3x+31)° and angle (2x-6)° are supplementary angles. Then the value of the variable x is 31°.
What is an angle?The angle is the distance between the intersecting lines or surfaces. The angle is also expressed in degrees. The angle is 360 degrees for one complete spin.
Supplementary angle - Two angles are said to be supplementary angles if their sum is 180 degrees.
Angle (3x+31)° and angle (2x-6)° are supplementary angles. Then we have
(3x + 31)° + (2x - 6)° = 180°
5x + 25° = 180°
5x = 155°
x = 31°
Then the value of the variable x is 31°.
Your question was incomplete, probably the question was...
Find the value of x. Angles (3x+31)° and (2x-6)° are supplementary angles.
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The value of x equals 31°.
The value of (x) can be found by solving the equation formed by the two angles in the given question:
Given angles:
Angle 1: [tex](3x + 31)^\circ[/tex]Angle 2: [tex](2x - 6)^\circ[/tex]Linear pair property: The sum of the measures of two angles forming a linear pair is (180°).
Now,
Set up the equation: (3x + 31) + (2x - 6) = 180
=> (3x + 2x) + (31 - 6) = 180
=> 5x + 25 = 180
Solve for (x): 5x = 155 => x = 31
Therefore, the value of (x) is 31°.
Disclaimer : I attached the image of complete question
_______ of functions are roots of polynomial equations. Zero(s) minimum(s) maximum(s) y-intercept(s)
Graph the following and determine in which quadrants the graphs lie: y= 1/3x
y = 1/3x
The slope is 1/3 ( the slope is the number with the variable x).
There is no other operation ( addition or subtraction) so the y- intercept is 0
The first point of the graph would be (0,0)
Calculate another point , lets use x = 3
y = 1/3(3) = 1
The second point is (3,1)
The line of the graph is an angled line which is located in quadrant 1 and 3.
See attached picture:
The formula for the perimeter of a rectangle is P=2(l+w).
Solve for w.
w=2P−l
w=p−l/2
w=p/2−l
w=P/2+l
The formula for 'w' in terms of the perimeter 'P' and length 'l' of the rectangle is w = (P - 2l)/2.
To solve for 'w' in the formula for the perimeter of a rectangle, P = 2(l + w), we need to isolate 'w' on one side of the equation.
P = 2(l + w)
First, distribute the 2 to both terms inside the parentheses:
P = 2l + 2w
Now, subtract 2l from both sides to move it to the other side of the equation:
P - 2l = 2w
Finally, divide both sides by 2 to solve for 'w':
w = (P - 2l)/2
Therefore, the formula for 'w' in terms of the perimeter 'P' and length 'l' of the rectangle is:
w = (P - 2l)/2
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Bartholomew just got a new credit card that offers both an introductory apr and a standard apr. If the standard apr. If t standard apr is 19.8% which of the following rates would most likely be the introductory apr?
A: 9.8%
B: 29.8%
C: 39.8%
D: 19.8%
Answer:
9.8
Step-by-step explanation:
Answer:
Its A. 9.8 on APEX hope this helps!!
Write the linear factorization of f(x)=x^4 - 2x^3 + 2x - 1
[tex]f(x)=x^4-2x^3+2x-1=x^4-x^3-x^3+x+x-1\\\\=x^3(x-1)-x(x^2-1)+1(x-1)\\\\=x^3(x-1)-x(x^2-1^2)+1(x-1)\\\\=x^3(x-1)-x(x+1)(x-1)+1(x-1)\\\\=x^3(x-1)+(x-1)(-x(x+1)+1)\\\\=(x-1)(x^3-x(x+1)+1)\\\\=(x-1)(x^3-x^2-x+1)\\\\=(x-1)[x^2(x-1)-1(x-1)]\\\\=(x-1)(x-1)(x^2-1)\\\\=(x-1)(x-1)(x^2-1^2)\\\\=(x-1)(x-1)(x-1)(x+1)[/tex]
[tex]Used:\\\\a^2-b^2=(a-b)(a+b)[/tex]
Find f(3) 1 point Multiple choice + Graph!
we know that
f(3) is the value of the function when x is equal to 3
so
For x=3
see in the graph
f(3)=6
see the attached figure
therefore
the answer is
f(3)=6
According to the graph in the problem, f(3) = 6
How to find the valueIn the graph we have curve and line. we look out for domain of 3
The domain of 3 is the point on the graph where x axis is 3. At this point we trace to the graph and find the corresponding value on the y axis
This gives the solution of the graph
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100 points please help!! The following data show the number of dogs per house in a neighborhood: 2, 1, 3, 2, 1, 4, 1, 5, 2, 1, 3, 1 The data have been organized into a dot plot, as shown below: A dot plot with integers 1 to 5 is shown. It is labeled Dogs, and titled Dogs in Houses. There are 4 dots above 1, 3 dots above 2, 2 dots above 3, 1 dot above 4, and 1 dot above 5. For which number are the data plotted incorrectly?
A. 4
B. 3
C. 2
D. 1
Hey there! The correct answer is D.1 the reason is because the number of the data is plotted incorrectly.
Kim is paid $78 for 6.5 hours of work. What is her rate of pay per hour?
Answer:
Answer:
78 dollars /6.5 hours = 12 dollars .
Your rate is 12 dollars per hour.
Step-by-step explanation:
please help asap 25 pts
What's the name of the line segment that passes through the center of a circle and has both endpoints on the circle?
The correct answer would be diameter. A chord is a line segment connecting two points on a circle's circumference. When the chord passes through the center of a circle it is called a diameter.
Final answer:
The line segment that passes through the center of a circle and has both endpoints on the circle is called a diameter.
Explanation:
The line segment which passes through the center of a circle and has both endpoints on the circle is called a diameter.
A diameter is the longest chord in a circle and it divides the circle into two equal halves. It passes through the center, and any point on the circumference of the circle is equidistant from the center.
For example, in the circle below, AB is a diameter:
Provide reasons for the proof.
Given: ∠1 ≅∠3 and
Prove: LT || MV
Proof:
Statement Reason
1. PV || QM Given
2. ∠1 is supplementary to ∠2 Same-Side Interior Angles Theorem
3. m∠1 + m∠2 = 180° Same-Side Interior Angle Theorem
4. ∠1 ≅ ∠3 Given
5. m∠1 = m∠3 Definition of Congruent angles => (If the angles are congruent, the measurements will be equal)
6. m∠3 + m∠2 = 180° Same side interior angle theorem (two angles on the same side make 180° if directly next to each other)
hope this helps
To prove that LT is parallel to MV, we can use the given information that angle 1 is congruent to angle 3. By showing congruence between triangles LTM and MVL, we can establish the proportionality of sides LT and MV, leading to the conclusion that LT is parallel to MV.
To prove that LT is parallel to MV, we need to use the given information that angle 1 is congruent to angle 3. Here's a step-by-step explanation:
Start by drawing a diagram representing the given information. Label the angles and the lines appropriately.Since angle 1 is congruent to angle 3, triangle LTM is congruent to triangle MVL by Angle-Angle-Angle (AAA) congruence.This means that corresponding sides of the triangles will be proportionate. The corresponding sides are LT and MV.By the Side-Side-Side (SSS) similarity theorem, we can conclude that LT is parallel to MV.Therefore, we have proved that LT is parallel to MV using the given information that angle 1 is congruent to angle 3.
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when simplifying an expression that contains no parentheses, you always multiply before you divide.
True
or
False
Answer:
I think the answer is false I am not A 100%sure but most likely false
Step-by-step explanation:
Two cars leave a gas station at the same time, one traveling north and the other south. The northbound car travels at 50 mph. After 3 hours the cars are 345 miles apart. How fast is the southbound car traveling?
a.
60 mph
b.
65 mph
c.
70 mph
d.
75 mph
For this case we have:
We define the following variable
x : Speed of the car in South direction
By definition, we know that:
[tex]Distance = Speed * Time[/tex]
For the car in north direction:
[tex]D1 = V1 * T1\\D1 = 50mph * 3h\\D1 = 150 miles[/tex]
For the car in south direction:
[tex]D2 = V2 * T2\\D2 = x * 3\\D2 = 3x[/tex]
The distance between both cars, after 3 hours, is:
[tex]D = D1 + D2[/tex]
So, we have:
[tex]345 = 150 + 3x[/tex]
Clearing x;
[tex]3x = 345-150\\3x = 195\\x = 65[/tex]
Thus, the speed of the car in the south direction is [tex]x = 65mph[/tex]
Answer:
Option B
Answer: B. 65 MPH
Step-by-step explanation:
Find a cubic function with the given zeros. (5 points)
-2, 5, -6
The cubic function with zeros -2, 5, and -6 is given by f(x) = (x + 2)(x - 5)(x + 6), derived from the zero-product property.
Explanation:To find a cubic function with the given zeros of -2, 5, and -6, we can use a formula associated with the zero-product property. This property states that if a product of factors equals zero, then at least one of the factors has to be equal to zero. So if -2, 5, and -6 are the zeros of the function, the factors of the function would be (x + 2), (x - 5), and (x + 6).
The cubic function f(x) is then derived from the multiplication of these factors. Therefore the cubic function with the zeros -2, 5, and -6 will be f(x) = (x + 2)(x - 5)(x + 6).
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40 people adopted and 30 more people adopted
An animal shelter currently has 27 dogs and 23 cats. What is the ratio of cats to animals
Answer:
Ratio of Cats to animals [tex]\frac{23}{50}[/tex]
Step-by-step explanation:
Total animals in the shelter = Total number of dogs + Total number of Cats
=[tex]27+23[/tex]
=[tex]50[/tex]
Ratio of Cats to animals = Number of Cats / Total animals
=[tex]\frac{23}{50}[/tex]
The sales tax in your city is 8.8%, and an item costs $63 before tax.How much tax would you pay on that item?Round to the nearest hundredth or cent.
please help on this one?
m = [tex]\frac{1}{2}[/tex], (x₁ , y₁) = (-2, -4)
y - y₁ = m(x - x₁)
y -(-4) = [tex]\frac{1}{2}[/tex](x - (-2))
y + 4 = [tex]\frac{1}{2}[/tex](x +2)
y + 4 = [tex]\frac{1}{2}x[/tex] + 1
y = [tex]\frac{1}{2}x[/tex] - 3
Answer: A
PLEASE HELP !! IT WOULD BE GREATLY APPREACIATED ! I DONT KNOW HOW TO DO THIS
(A) Bella's TV ratio of sides: 36"/27" = 1.333...
Lenny's: 52"/29.25" = 1.777...
Bella's and Lenny's TVs are NOT similar - their side ratios are different. Bella's is 4:3 while Lenny's is close to 16:9.
(B) Bella's TV has an exact ratio of 4:3 (36/27=4/3) and so is the older type.
Lenny's TV has a ratio of 52"/29.25". Divide both 52 and 29.25 by 3.25 to get 16/9, or 16:9. So Lenny's is the newer type.
What is the slope of the line represented by -14y = 7x ?
I believe the answer is -1/2.
A cylindrical tank is half full of oil. The cylinder has a base radius of 100cm. The height of the cylinder is 250cm. 1litre=1000cm How many litres of oil are in the tank
≈ 3927 litres
the volume (V) of a cylinder is calculated using the formula
V = πr²h ( r is the radius of base and h is the height )
V= π × 100² × 250 = 7853981.634 cm³
[tex]\frac{1}{2}[/tex] V = 3926990.817 = [tex]\frac{3926990.817}{1000}[/tex] ≈ 3927 litres
968,000 in Scientific notation with exponents
Answer:
9.68E5
Step-by-step explanation:
The table below represents the relationship of the amount of snowfall ( in inches) in 5 cities to the amount of time
(in hours) of a recent winter storm.
Answer:
its c
Step-by-step explanation:
Given: △MOP, Perimeter of △MOP=12+4 sqr rt. 3 m∠P=90°, m∠M=60° Find: MP
Consider triangle MOP. This triangle is right triangle, because m∠P=90°.
Since m∠P=90°, m∠M=60°, then m∠O=180°-90°-60°=30°.
The leg opposite to the angle with measure 30° is half of the hypotenuse. Let the leg MP=x, then the hypotenuse MO=2x. By the Pythagorean theorem,
[tex]PO^2+MP^2=MO^2\Rightarrow PO^2=(2x)^2-x^2=4x^2-x^2=3x^2\Rightarrow PO=\sqrt{3}x.[/tex]
The perimeter of triangle MOP is
[tex]P_{MOP}=MO+PO+MP=x+2x+\sqrt{3}x=3x+\sqrt{3}x.[/tex]
Since [tex]P_{MOP}=12+4\sqrt{3},[/tex] then [tex]x=4.[/tex]
Answer: MP=4 units.
Answer:
mp=4 cuz in a 30-60-90 degree right triangle if the shorter leg is x, then the longer leg (po) is xsqrt3. Because the perimeter is 12+4sqrt3, that means the longer leg is probably 4sqrt3. because of this, then the shorter leg, mp is 4. Since the hypotenuse is equal to 2x, then it would be 2*4=8=PO.
During a promotional event a sporting goods store gave a free t-shrit to every 8th customer and a free water bottle to every 10th customer which customer has the first to get a free t-shirt and a free water bottle
Given
During a promotional event
A sporting goods store gave a free t-shrit to every 8th customer and a free water bottle to every 10th customer.
Find out which customer has the first to get a free t-shirt and a free water bottle.
To proof
Definition of LCM ( Least common multiple )
The LCM of two numbers is the smallest number that they both divide evenly into.
As given in the equation
A sporting goods store gave a free t-shrit to every 8th customer and a free water bottle to every 10th customer.
Let take 8 be as one number and 10 be as another number.
LCM ( 8,10 ) = 2× 2 ×2 ×5
= 40
Also
A sporting goods store gave a free t-shrit to every 8th customer and a free water bottle to every 10th customer.
This is shown in the table given below.
As shown in the table the common value is 40.
Therefore
The first to get a free t-shirt and a free water bottle is 40th customer.
Hence proved
Answer:
The 40th customer was the first to get a free t-shirt and free water bottle
Step-by-step explanation:
Factor the expression completely over the complex numbers.
x^4 − 625
Enter your answer in the box.
You have to observe that [tex] x^4 [/tex] and 625 are both squares (of [tex] x^2 [/tex] and 25, respectively). So, you can use the "difference of square" pattern for factorization:
[tex] a^2-b^2 = (a+b)(a-b) [/tex]
to write
[tex] x^4 - 625 = (x^2+25)(x^2-25) [/tex]
Note that, again, [tex]x^2-25[/tex] is a difference of square:
[tex] x^2-25 = (x+5)(x-5) [/tex]
On the other hands, [tex] x^2+25 [/tex] admits no factorization, because it's a second-degree polynomial (thus a parabola) with no solutions.
So, the whole expression becomes
[tex] x^4 - 625 = (x^2+25)(x+5)(x-5) [/tex]
Which angle corresponds to <7?
A. <1
B. <3
C. <4
D. <6
Based on the parallel lines, an angle that corresponds to ∠7 include the following: B. ∠3.
What are corresponding angles?In Mathematics and Geometry, corresponding angles theorem is a theorem which states that corresponding angles are always congruent when the transversal intersects two parallel lines:
This ultimately implies that, the corresponding angles will be always equal (congruent) when a transversal intersects two parallel lines.
By applying corresponding angles theorem to the two parallel lines, we have the following congruent angles:
m∠3 ≅ m∠7
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What is the product? Enter your answer as a fraction, in simplified form, in the box. −4/5⋅10/16
Answer:
-1/2
Step-by-step explanation:
-4 x 10= -40
-5 x 16= -80
-40 is half of -80, so the fraction would be -1/2
Hope this helps! <3
The product of the given expression of a fraction is -1/2.
It is required to find the product.
What is fraction?A fraction is a part of a whole. In arithmetic, the number is expressed as a quotient, in which the numerator is divided by the denominator. In a simple fraction, both are integers. A complex fraction has a fraction in the numerator or denominator. In a proper fraction, the numerator is less than the denominator.
Given:
The product of the given expression of fractions in simplified form.
According to given question we have,
[tex]\frac{-4}{5} *\frac{10}{16} \\\\=\frac{-40}{80} \\\\=\frac{-1}{2}[/tex]
so, the fraction would be -1/2.
Therefore, the product of the given expression of a fraction is -1/2.
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