If ɑ and β are the roots of x² + bx + c = 0, then we can write
x² + bx + c = (x - ɑ) (x - β)
Expanding the right side gives
x² + bx + c = x² - (ɑ + β) x + ɑβ
so that ɑ + β = -b and ɑβ = c.
Recall that for all real numbers m and n,
(m + n)² = m² + 2mn + n²
a) It follows that
(i) ɑ² + β² = (ɑ + β)² - 2ɑβ = (-b)² + c = b² + c
(ii) (ɑ - β)² = ɑ² - 2ɑβ + β² = b² + c - 2c = b² - c
b) I assume you mean to find the quadratic whose roots are ɑ² + β² and (ɑ - β)² (and not (ɑ - 3)²). The simplest quadratic of this form is
(x - (ɑ² + β²)) (x - (ɑ - β)²)
Using the results from part (a), this becomes
(x - (b² + c)) (x - (b² - c))
and expanding, we get
x² - (b² + c + b² - c) x + (b² + c) (b² - c)
= x² - 2b² x + b⁴ - c²
Find the slope of the line that passes through the points f(4) = 6 and f(-2) = 8.
Answer:
slope = - [tex]\frac{1}{3}[/tex]
Step-by-step explanation:
given f(4) = 6 and f(- 2) = 8 , then 2 points on the line are
(4, 6 ) and (- 2, 8 )
calculate slope m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (4, 6 ) and (x₂, y₂ ) = (- 2, 8 )
m = [tex]\frac{8-6}{-2-4}[/tex] = [tex]\frac{2}{-6}[/tex] = - [tex]\frac{1}{3}[/tex]
Solve the formula for a linear equation, y = mx + b, for m. I’m not so great at formulas and it’d be great if someone helped thank you :)
Answer:
See below
Step-by-step explanation:
mx + b = y subtract 'b' from both sides of the equation
mx = y - b now, divide both sides by 'x'
m = (y-b)/x
The sun of two numbers is 7 and a different 1 what are the two numbers
Answer:
3 and 4
Step-by-step explanation:
given that the difference is 7, two numbers that add up to 7 are 3 and 4.
A rectangular piece of iron has sides with lengths of 7.08 × 10–3 m, 2.18 × 10–2 m, and 4.51 × 10–3 m. what is the volume of the piece of iron? 6.96 × 10–7 m3 6.96 × 107 m3 6.96 × 10–18 m3
The volume of the rectangular piece with sides of lengths of 7.08 × 10–3 m, 2.18 × 10–2 m, and 4.51 × 10–3m is 6. 96 × 10^ -7 m
What is volume?Volume is defined as the amount of space or capacity occupied by a three-dimensional shape.
It is represented in cubic units.
How to determine the volume of a rectangleFormula:
Volume, V = length × breadth × height
Given the values;
7. 08 × 10 ^-3m , 2.18 × 10^ -2m and 4. 51 × 10^-3m
Multiply all the lengths
Volume = 7. 08 × 10 ^-3 × 2.18 × 10^ -2 × 4. 51 × 10^-3
= 69.6 × 10^-8
= 6. 96 × 10^ -7 m
Therefore, the volume of the rectangular piece is 6. 96 × 10^ -7 m
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Answer:
A
Step-by-step explanation:
trust
Find the minimum point(s) of f(x) = x - 2x² + 1 given that f'(x) = 4x³ - 4x.
Answer: f(x) = x - 2x² + 1
x=-b/2a
(1/4,9/8
2. f'(x) = 4x³ - 4x
((\sqrt(3))/(3),-(8\sqrt(3))/(9))is a local minima
(-(\sqrt(3))/(3),(8\sqrt(3))/(9))is a local maxima
Step-by-step explanation: hopes this help
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Jayne Stopped to get gas before going on a road trip the tank already had 4 gallons of gas in it which equation relates the total amount of gasoline in the tank y to the number of gallons that she put in the tank X
a.) y=4+x
b.) y = x - 4
c.) y = 4 . x
d.) y = x / 4
Answer: A.) y=4+x
Step-by-step explanation:
What is the first step in the solution of 3x2+6x=12 by completing the square?
A: r3+r=2
B: r2+2r=4
C: 3r2+6r+9=12+9
D: 3r2+6r+36=12+36
Answer:
B: r² + 2r = 4
Explanation:
[tex]\rightarrow \sf 3x^2+6x=12[/tex]
divide both sides by 3
[tex]\rightarrow \sf x^2+2x=4[/tex] ← This is the result of the first step.
factor out the common term
[tex]\rightarrow \sf x(x+2) = 4[/tex]
set to 0
[tex]\rightarrow \sf x = 4, \ x+2 = 4[/tex]
simplify
[tex]\rightarrow \sf x = 4, \ x= 2[/tex]
Divide all terms by a i.e 3
We get
x²+2x=4Or
r²+2r=4Option B is correct
evaluate the exppresion 14x-11y if x=4 and y=3
PLEASE HELP ILL MAKE YOU THE BRAINIEST what is the area of the regular polygon
The area of a regular polygon is one-half the product of its apothem and its perimeter. Often the formula is written like this: Area=1/2(ap), where a denotes the length of an apothem, and p denotes the perimeter.
Step-by-step explanation:
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Denise is designing the seating arrangement for a concert at an outdoor theater. to give everyone a good view, each row must have 6 more seats than the row before it, and the 1st row can only have 11 seats. explain to denise how to create an equation to predict the number of seats in any row. then, using your equation, show your work to determine the number of seats in row 18
An equation is formed of two equal expressions. The number of seats in row 18 will be 119.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
A.) Given the number of seats that the first row can have is 11, while the number of seats that the next row should have is six more than the previous one. Therefore, the number of seats will increase linearly.
Thus, the equation for the number of seats in any row can be given as y=11+6x, where y is the number of seats and x is the number of row.
B.) The number of seats in row 18 will be equal to,
y = 11 + 6x
y = 11 + 6(18)
y = 11 + 108
y = 119
Hence, the number of seats in row 18 will be 119.
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Anthony shines a beam of light on the surface of a solution of glycerol at a certain angle. if the angle of refraction is 38°, what is the angle of incidence? the index of refraction of glycerol is 1.47, while that of air is 1.00. a. 38° b. 45° c. 55° d. 65° e. 70°
Answer:
D) 65°
Step-by-step explanation:
Use Snell's Law
[tex]n_1\sin\theta_1=n_2\sin\theta_2\\\\1.00\sin\theta_1=1.47\sin38^\circ\\\\\sin\theta_1=1.47\sin38^\circ\\\\\theta_1=\sin^{-1}(1.47\sin38^\circ)\\\\\theta_1\approx64.83^\circ[/tex]
Thus, the angle of incidence is D) 65°
Help, I still don't understand this question!
What is the measure of ∠BAC, ∠ABC, and ∠ACB?
The answer is:
" m∠BAC = 45° ; m∠ABC, = 64° ; and: m∠ACB = 70° ."
__________
Step-by-step explanation:
To find the measure of ∠BCA {"m∠BCA"} :
<BCA is supplementary to " 110 " ; that is, both, or all, of all the measurements that form a "line or line segment"; or any portions thereof,
that can be written in: "slope-intercept format"
that is, on Cartesian plane: " y = mx + b " ;
in which: "y" is isolated on the "left-hand side" of the equation; as a single, isolated variable;
and in which: "m" is the "slope" (if applicable); and the coefficient to the "x" value'
and in which "b" is the "y-intercept"; or more precisely"
the value of the "y-coordinate"—in: "(x, y)" format —at which:
" x = 0" ; that is, in the format: " (0, b)."
_________________
So, to find
m∠BCA : Subtract our given value, "110" ; from "180" ; as follows:
_________________
" (180 − 110) = 70" ;
__________________
Now: Note a triangle, by definition, has 3 (three) sides and 3 (three) angles—and all three (3) angles of a triangle add up to 180°.
So: We just solved for m∠ACB ; which is: 70° .
Now shall solve for the value of "x" ; & then find the value of all 3 (three) sides of this triangle
Since all angles within a triangle add up to 180: Let's add them up:
" 70 + 3x + 20 + 3x = 180 " ;
Combine the "like terms" on the "left-hand side" of the equation:
" +3x + 3x = 6x " ; "+ 70 + 20 = 90 ";
And rewrite the equation:
" 6x + 90 = 180 " ;
Now, subtract "90" from Each Side of the equation:
" 6x + 90 − 90 = 180 − 90 " ;
to get: " 6x = 90 " ;
Now, divide Each side of the equation by "6" ;
to isolate "x" on one side of the equation ;
and to solve for "x" :
" 6x / 6 = 90 / 6 ;
to get: " x = 15 " ;
________________
Now, we are asked to find:
1) " m∠BAC = 3x " ; Plug in "15" for "x" ; and solve:
"3x = 3(15) = 45 " ;
2) " m∠ABC = "(3x + 20)" ; Plug in "15" for "x"; and solve:
"3(15) + 20 = 45 + 20 = 65 " ;
3) " m∠ACB = 70 " ; (as per our calculated value);
To check our work—do these 3 (three) numbers add up to 180?
"45 + 65 + 70 =? 180 ? " ;
" 110 + 70 = " 180 "? Yes!
__________
So: The answer is as follows:
" m∠BAC = 45° ; m∠ABC, = 64° ; and: m∠ACB = 70° ."
__________
Hope this is helpful to you! Best wishes!
elo guys can u guys help me on these i have 20 min
I used exemplar. the mean means add all the numbers and divide it buy how many there are.
5 + 7 + 9 = 21
21/3 = 7
0 3 4 6 6 7 7 7 9
Median = 6 (the middle number)
Mode = 7 (the most common number)
Range = 9 (Biggest number - smallest number)
Hope this helps!
If not then let me know!
Nevaeh models the volume of a popcorn box as a right rectangular prism. Its dimensions are 5 in by 3 in by 7 1/4 in. How many cubic inches of popcorn would it hold when it’s a quarter full? Round your answer to the nearest tenth if necessary.
The cuboid can hold 27.2 cubic inches of popcorn
What is a right rectangular prism?A right rectangular prism is also called the cuboid. it is a solid three dimensional shape with 6 faces, 12 edges and 8 vertices.
Analysis:
Volume of cuboid = length x width x height
= 5 x 3 x 29/4 = 108.75 cubic inches
When it is quarter full, volume = 1/4 x 108.75 = 27.2 cubic inches
In conclusion, the volume of the cuboid is 27.2 cubic inches.
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In a certain week, the time Lisa spent on
sport was 3 hours less than twice the time
she spent doing her school homework. The
time she spent on reading was of the time
she spent doing her school homework. The
total time she spent on these three
activities was 30 hours in that week. Form
an equation and solve it to find the amount
Answer:
2M + 3 hrs
Step-by-step explanation:
Lisa spent doing her mathematics homework = M hrs
the amount of time spent watching television was 3 hours more than twice the time she spent doing her mathematics homework.
lisa spent on watching television = 2M + 3 hrs
Total time spent on these two activities = M + 2M + 3
= 3M + 3 hrs
3 M + 3 = 30
=> 3M = 27
M = 9
-33 + 8x = -3 (3x+6) + 2
Answer:
x = 1
Step-by-step explanation:
-33 + 8x = -9x - 18 + 2
8x+ 9x = -18 + 2 + 33
17x = -16 + 33
17x = 17
x = 1
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Find the measure of the missing angle.
112°
a
The measure of angle b is 112° and the measure of angle c is 68°
Calculating the measures of anglesFrom the question, we are to determine the measures of the missing angles
From the diagram,
b = 112° (Vertically opposite angles)
∴ b = 112°
Also,
112° + c = 180° (Sum of angles on a straight line)
c = 180°- 122°
∴ c = 68°
Hence, the measure of angle b is 112° and the measure of angle c is 68°
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An investment of $2,375 is made in a 7-year cd that earns 314% annual interest that is compounded continuously. how much will the cd be worth at the end of the 7-year term? round your answer to the nearest cent. do not include the dollar sign.
The worth of the cd at the end of 7 years will be $384,101.79.
What will be the worth of the investment?When an investment earns an interest that is compounded continuously, it means that it earns interest continuously.
The formula for calculating future value when there is continuous compounding is : A x e^r x N
Where:
A= amount e = 2.7182818 N = number of years r = interest rate2375 x (e^3.14) x 7 = $384,101.79
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write Six times a number w is at least -24 as an inequality
Answer:
[tex]6n \geq -24[/tex]
Step-by-step explanation:
First let's provide a variable for "a number" and call it [tex]n[/tex].
So "six times a number" could be written as [tex]6*n[/tex].
"Is at least" in mathematical terms means "greater than or equal to" or
≥
Putting this altogether gives the expression:
[tex]6n\geq -24[/tex]
REASONING A rectangular box has dimensions of x inches by (x+5) inches by (x−2) inches. The volume of the box is 30x cubic inches. Find the dimensions of the box. Write the dimensions in ascending order.
I need help, please. I don’t understand
Answer:
60%
Step-by-step explanation:
So, this question is saying the bowls can be between a diameter of 6.95 and 7.05
Box 1, 3 and 5 are in this margin.
Box 2 and 4 are not.
3/5 boxes can be included.
3/5 is 6/10
6/10 is 60%
A cone has a height of 9 feet and a radius of 8 feet. What is its volume?
Use ≈ 3.14 and round your answer to the nearest hundredth.
please help me i dont get it. :(
Help ASAP
Ben is making a chart in which he is listing the masses of different planetary bodies. He plans to write the masses in scientific notation. Convert the values in scientific notation to standard notation.
The mass of Ganymede is 1.48E23, or , kilograms.
The mass of Pandora is 2.20E17, or , kilograms.
Step-by-step explanation:
I think 2.20e23 because it said bodies not body
Change 24 inches into feet given the fact that 1 foot = 12 inches.
Answer:
2 feet
Step-by-step explanation:
1 foot = 12 inches
2 feet=24 inches
Answer:
2 feet
Step-by-step explanation:
12 x 2 is 24.
can someone help me with this question
Answer:
Step-by-step explanation:
Proof
AB||CD Given
<1 = <2 Given
<2 = <3 Alternate interior angles of Parallel lines
<1 = <3 Substitution for <2 from Given.
AC = AB the sides opposite equal angles are equal
PLEASE HELP!!!!!!!!!
Answer:
CE = 4.6 ft
Step-by-step explanation:
(a)
given 2 secants drawn from an external point to the circle , then
the product of the external part of one secant and the entire secant is equal to the product of the other secant's external part and that entire secant, that is
BE × BC = AB × DB
(b)
substituting given values into the equation
BE × 10 = 3 × 18
10 BE = 54 ( divide both sides by 10 )
BE = 5.4 ft
then
CE = CB - BE = 10 - 5.4 = 4.6 ft
f(x) = x^2. What is g(x)?
the answer is XG and that is the answer to your question ❓ ok she will he hi she
Explanation:
Normally x = 1 maps to y = f(x) = 1 for the parent function f(x) = x^2
However, the red curve shows x = 1 leads to y = 16 instead. This is a jump of "times 16"
We must multiply the f(x) outputs by 16 to get the g(x) outputs.
g(x) = 16*f(x)
g(x) = 16x^2
IM SO LOST PLS HELP ME
[tex]~~~~~~~c = \dfrac{a^2 +3b}{4}\\\\\implies 4c = a^2 +3b\\\\\implies 3b = 4c-a^2\\\\\implies b = \dfrac{4c-a^2}{3}[/tex]
Answer:
b = [tex]\frac{(4c-a^2)}{3}[/tex]
Step-by-step explanation:
[tex]c=\frac{a^{2}+3b }{4}[/tex]
First, let us multiply both sides by 4.
4c = a² + 3b
Now let us take a² to the left side.
4c - a² = 3b
Now to make b as the subject let us divide both sides by 3.
[tex]\frac{(4c-a^2)}{3} =\frac{3b}{3}[/tex]
Therefore,
b = [tex]\frac{(4c-a^2)}{3}[/tex]
Help me if you can please!!!
To solve this, we need to use our trigonometric equations:
⇒ look at the diagram attached
Let's examine what the problem wants:
the length of the side opposite to the 27-degree angleLet's examine what the problem gives us:
the length of the side adjacent to the 27-degree anglethe angle that has a measure of 27-degreesLet's solve:
[tex]tan(27)=\frac{x}{15} \\x=15*tan(27)\\x=7.64[/tex]
Answer: 7.6 (rounded to the nearest tenth)
Hope that helps!
Round of to the nearest thousand 741,543
Answer:
The correct answer is 742,000
Step-by-step explanation:
Answer:
742,000
Step-by-step explanation:
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