Answer:First suppose that the roots of the equation
x2−bx+c=0(1)
are real and positive. From the quadratic formula, we see that the roots of (1) are of the form
b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2.
For the root or roots to be real, we require that b2−4c≥0, that is, b2≥4c. In order for them to be positive, we require that
b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0.
This immediately tells us that b>0, but we can go further. We can rearrange this to get
b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,
which (assuming that b>0) is true if and only if
b2>b2−4c,
since both sides of the inequality are positive so we may square. But then
4c>0.
That is, if the roots are real and positive then b>0 and b2≥4c>0.
Now suppose that b>0 and b2≥4c>0.
Then the roots of (1) are real since b2−4c≥0, and b>0 guarantees that the root b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√ is positive.
So it remains to show that b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√>0. We have that
4c>0,
so that
b2>b2−4c,
then square rooting shows that
b>b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√,
so the roots of (1) are real and positive, as required.
Approach 2
the curve y equals x squared minus b x plus c showing two positive roots for y equals zero
This is intended to be a proof without words! We have from the diagram that:
If c, b and b2−4c are all positive, there are two real positive roots for x2−bx+c=0 (if b2=4c, we have two real positive equal roots).
If there are two real positive roots for x2−bx+c=0, then c and b are positive and b2−4c is non-negative.
(Why is the distance between the roots b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√?)
Approach 3
When solving problems about the roots of polynomials, it is often useful to find expressions those roots must satisfy and see if this tells us anything new. If α and β denote the roots of the equation, then
x2−bx+c=(x−α)(x−β)=x2−(α+β)x+αβ
and so α+β=b and αβ=c.
We also know that the roots of a quadratic equation are real if and only if the discriminant is non-negative, that is, if and only if b2−4c≥0.
Using these facts, if α and β are both real and positive, then b=α+β>0, c=αβ>0 and b2≥4c, as above.
Conversely, if b>0 and b2≥4c>0, then we know the discriminant is positive and hence both roots are real. We also have that
αβ>0(2)
and
α+β>0.(3)
As α and β are both real, by (2), we know that α and β are either both positive or both negative. However, if α and β were both negative, then (3) could not possibly hold. Hence α and β are both positive, as required.
We now sketch on a graph the region where b>0, c>0 and b2≥4c:
The curve b squared = 4 c as a quadratic with the c-axis vertical and the b-axis horizontal. The region below it is shaded.
The region of the b-c plane for which b>0, c>0 and b2≥4c
Sketch the region of the b-c plane in which the roots of the equation are real and less than 1 in magnitude.
We know that in order for the roots to be real we need b2≥4c as in the first part. We now need to find the region where
−1<b±b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.(4)
We have b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2≤b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2 so we only need to consider the values for which both
−1<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2andb+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√2<1.
Firstly, we will consider the values for which
b+b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2.
Rearranging gives
b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<2−b.
So b<2, as the square root is non-negative, and we can square both sides to get
b2−4c<4−4b+b2,
which we may rearrange to find c>b−1.
We will now consider the values for which
−2<b−b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√.
Similarly, we can rearrange to get
b2−4c⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯√<b+2.
So b>−2, and we can, as before, square to get
b2−4c<b2+4b+4,
and hence c>−b−1.
To sketch the graph, we start by considering the boundary curves b2=4c, c=b−1 and c=−b−1, and the points at which they intersect. We can see that the two lines only intersect when b=0 and c=−1, and the lines intersect the curve when
b2=4b−4andb2=−4b−4
which rearrange to
(b−2)2=0and(b+2)2=0.
This tells us that each line intersects with the curve in only one place and so these lines must be tangent.
Is there a way we could have deduced this directly from (4)?
What do the lines being tangent signify in terms of our equation x2−bx+c=0?
Is this a representation of a well-known property of these equations?
Sketching the graph then yields the following picture:
The graph with the previous curve and the lines 4 c = 4 b minus 4 and 4 c = minus 4 b - 4. Each line touches the curve once and the two lines intersect at (0, minus 1). The region between the three lines/curves is shaded.
The shaded region is where b2≥4c, c>b−1, c>−b−1 and −2<b<2
We might notice that when we derived the inequalities c>b−1 and c>−b−1, a lot of the work we did was very similar.
We might also notice that the graph above is symmetric about the c axis. Why is this?
Does the graph give us any ideas about how we might deduce one inequality from the other?
Step-by-step explanation:
(-8/9) divided by (-2/3)x(-4 1/2)
Answer:
-0.29629629629 or -0.3
Step-by-step explanation:
first do (-2/3)*(-4 1/2) which is 3
then divide (-8/9) by 3 and you get -0.29629629629 which can be rounded to -0.3
The Answer will be -8 using fraction operation.
What is a fraction?If the numerator is bigger, it is referred to as an improper fraction and can also be expressed as a mixed number, which is a whole-number quotient with a proper-fraction remainder.
Any fraction can be expressed in decimal form by dividing it by its denominator. One or more digits may continue to repeat indefinitely or the result may come to a stop at some point.
First solve the 2nd part which is
[tex]\frac{-2}{3}*-4\frac{1}{2} \\=3[/tex]
Now -8/3 * 3 = -8
Hence the answer for the given question will be -8.
Learn more about Fraction, by the following link.
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If you have 2\10 of an apple, how many fifths do you have
Answer:
4/5
Step-by-step explanation:
1 - 2/10 = 8/10 = 4/5
Answer:
[tex]\frac{1}{5}[/tex]
Step-by-step explanation:
You have [tex]\frac{2}{10}[/tex] ( divide numerator and denominator by 2 ), that is
[tex]\frac{2}{10}[/tex] = [tex]\frac{1}{5}[/tex]
If the side opposite the 60 degree angle in a 30-60-90 degree is 12 centimetres, then what length of the side opposite the 30 degree angle?
Answer:
4*[tex]\sqrt3[/tex]
Step-by-step explanation:
What is the first step in evaluating the expression shown below?
(12.9-3.1) 6.2-2 + 43
Answer:
Subtract 3.1 from 12.9
Hope it helped
: What is the image after (3,7) is reflected over the x-axis?
Brainly
Got the question wrong cause the other answer here is the correct answer directly from Delta Math.
(-7, -3)
Enjoy and Happy Holidays!
2. Michael bought a sandwich that cost $4.69, a bag of chips that cost $1.99
and an iced tea for $2.50. How much did he spend in all? *
A. 7.19
B. 4.49
C. 6.68
D. 9.18
Answer:
9.18
Step-by-step explanation:
4.69
1.99
+2.50
9.18
At a real estate agency, an agent sold a house for $397,000. The commission rate is 4.5% for the real estate agency and the commission rate for the agent is 20% of the amount the real estate agency gets. How much did the agency make on the house? How much did the agent earn in commission?
Answer:
1. $97,265
2. $17,865
Step-by-step explanation:
Step one:
given data
cost of the house= $397,000.
we are told that the commission is 4.5% real estate agency
and also the agent commission is 20%
Step two:
1. How much did the agency make on the house?
this amount is 20% plus 4.5% together.
=20/100*397,000 +4.5/100*397,000
=0.2*397,000. +0.045*397,000.
=79400+17865
=97,265
The agent made a total of $97,265 on the house
2. How much did the agent earn in commission?
in commission, the rate is 4.5% which amounts to $17,865
La mitad de la edad que tenia anahí hace 27 años equivale a la cuarta parte de la edad que tendrá dentro de 5 años. ¿ qué edad tiene actualmente anahí?
Responder:
59 años
Explicación paso a paso:
Asumir que:
Edad actual = x
Edad 27 años atrás = x - 27
Edad dentro de 5 años = x + 5
La mitad de la edad de hace 27 años = 1/2 (x - 27)
Cuarto de edad dentro de cinco años = 1/4 (x +5)
0,5 (x - 27) = 0,25 (x + 5)
0.5x - 13.5 = 0.25x + 1.25
0.5x - 0.25x = 1.25 + 13.5
0,25 veces = 14,75
Dividir por 0,25
0,25x / 0,25 = 14,75 / 0,25
x = 59
Por lo tanto, la edad de Anahí ahora es de 59 años.
La edad actual de Anahí es de 59 años.
Sabemos por la pregunta que la mitad de la edad que tenía Anahí hace 27 años equivale a la cuarta parte de la edad que tendrá dentro de 5 años.
Para escribir esto como una ecuación, sea y la edad actual;
1/2(s - 27) = 1/4(y + 5)
El MCM de 2 y 4 es 4 por lo tanto;
4/2(y - 27) = 4/4(y + 5)
2(s - 27) =(y + 5)
2y - 54 = y + 5
2y -y = 5 + 54
y = 59
La edad actual de Anahí es de 59 años.
Más información sobre ecuaciones: https://brainly.com/question/2263981
solve the system by substitution -x+3y=15
y=2x
Answer:
(3,6)
Step-by-step explanation:
ratio// please help!!
Answer:
The loose sweets
Step-by-step explanation:
100 divided by 0.89 = 112
120 divided by 1.49 = 80
Answer:
Step-by-step explanation:
1.49 ÷ 120 = 0.0124 GBP per gram
0.89 ÷ 100 = 0.0089 GBP per gram < 0.01 (The price of loose sweets is better value)
HELP At the end of a snow storm, Eli saw there was a lot of snow on his front lawn. The
temperature increased and the snow began to melt at a steady rate. The depth of
snow on Eli's lawn, in inches, can be modeled by the equation S = 16 - t, where t
is the time, in hours, after the snow stopped falling. What is the y-intercept of the
equation and what is its interpretation in the context of the problem?
The y-intercept of the function is
which represents
Answer:
The y-intercept is 16, which represents the original depth of snow on the lawn
Step-by-step explanation:
t is the time in hours where the snow melted, so 16 was the original amount of snow on the ground before it melted over time.
The y-intercept is 16 inches. Then the y-intercept represents the height of the snow initially.
What is a linear equation?A connection between a number of variables results in a linear model when a graph is displayed. The variable will have a degree of one.
The linear equation is given as,
y = mx + c
Where m is the slope of the line and c is the y-intercept of the line.
Eli noticed a lot of snow on his front lawn towards the conclusion of a snowstorm. As the weather rose, the snow began to melt at a steady pace. The depth of snow on Eli's lawn in inches may be calculated using the equation S = 16 - t, where t is the time after the snow ceased falling in hours.
Compare the equation with the standard equation, then we have
c = 16
The y-intercept represents the height of the snow initially.
More about the linear equation link is given below.
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can someone please help me with this please and thank you
3 out of 4 students in one school named Apple
Pie as their favorite Thanksgiving food. If the
school has 100 students, how many chose Apple
Pie?
Answer
the answer is 75
Step-by-step explanation:
100 divided by 3 = 75 so there is 75 students
75 is the answer I have the same question as you
Which pair of complex numbers has a real-number product?
(1 + 31 (6)
(1 + 3)(2-31)
(1 + 3)(1 – 31)
0 (1 + 3/)(31)
Answer: C. (1 + 3i)(1 - 3i)
Step-by-step explanation:
Took the test on edgnuity
Answer:
The person ontop is correct! It is C.
Investors buy a studio apartment for $180,000. Of this amount, they have a down payment of $36,000. Their down payment is what percent of the purchase price? What percent of the purchase price would a $72,000 down payment be?
Their down payment is *blank*% of the purchase price.
Answer:
36,000 would be 20 percent. 72,000 would be 40 percent.
Step-by-step explanation:
You just divide 36,000 by 180,000
Evaluate each value expression for the given value of the variable : x+x+5 when x=7
QUICK PLSSSSSS
Answer:
19
Step-by-step explanation:
x = 7, so we have:
7 + 7 = 14
14 + 5 = 19
Answer:
19
Step-by-step explanation:
you replace the x with a seven. so it will look like
7+7+5= 19
number 10?? please help.
Answer:
∠GFM= 124°
Step-by-step explanation:
∠GFH= 180° -90° -(5x +9)° (∠ sum of △FGH)
∠GFH= 90° -5x° -9°
∠GFH= (81 -5x)°
∠GFM +∠GFH= 180° (adj. ∠s on a str. line)
25x -1 +81 -5x= 180
20x +80= 180 (simplify)
20x= 180 -80 (-80 on both sides)
20x= 100
x= 100 ÷20 (÷20 on both sides)
x= 5
∴∠GFM
= 25x -1
= 25(5) -1
= 124°
Please help it’s due in an hour! It’s algebra.
Answer:
You might wanna double check some of these because i might've forgot to flip some of the < and > signs but I'm pretty sure its correct
1) y = x < -18 and y = x < 18
2) y = -3 < x < 3
3) x = 1 < y < 6 (it just gets simplified)
4) x = -3 > y > -18
5) y = -2 > x and y = x < 3.5
6) y = 8.5 > x > 5.5
7) y = -5.75 < x < -3.5
8) y = 1 > x > 2.25
9) -17 < x < -5
Step-by-step explanation:
Which pair of complex numbers has a real-number product?
O (1 + 2i) (8i)
0 (1 + 2i) (2 - 5i)
O (1 + 2i) (1 – 2i)
O (1 + 2i)(4i)
Answer:
(1 + 2i)(1 - 2i)
Step-by-step explanation:
Multiplying a complex number and its conjugate results in a real number.
Thus
1 + 2i and 1 - 2i are conjugates
(1 + 2i)(1 - 2i) ← expand using FOIL
= 1 - 2I + 2I - 4I² [ note i² = - 1 ]
= 1 - 4(- 1)
= 1 + 4
= 5 ← a real number
Answer:
C
Step-by-step explanation:
C
1l) HAO WENT TO A HOBBY SHOP AND SAW A DISPLAY OF TOY TRAINS. THE SIGN IN FRONT OF
THE DISPLAY SAID THAT THE TOY TRAINS WERE A REPLICA OF THE LONG ISLAND RAILROAD
TRAINS. HAO ESTIMATED THAT ONE OF THE RAILROAD CARS WAS APPROXIMATELY 15
INCHES LONG AND 4 INCHES WIDE. THE SIGN IN FRONT OF THE DISPLAY STATED
"SCALE- 2 /9".
WHAT IS THE LENGTH OF THE RAILROAD CAR ON HE LONG ISLAND RAILROAD?
WHAT IS THE WIDTH OF THE LONG ISLAND RAILROAD CAR?
Answer:
Length = [tex]57.5\ \text{inches}[/tex]
Width = [tex]18\ \text{inches}[/tex]
Step-by-step explanation:
The scale of the toy train is [tex]\dfrac{2}{9}[/tex]
So, the dimensions of the toy train is [tex]\dfrac{2}{9}[/tex] times the actual dimensions of the train.
The dimensions of actual train is [tex]\dfrac{1}{\frac{2}{9}}=\frac{9}{2}[/tex] times the toy train's dimensions.
Length
[tex]15\times \dfrac{9}{2}=57.5\ \text{inches}[/tex]
Width
[tex]4\times \dfrac{9}{2}=18\ \text{inches}[/tex]
The length of the railroad car on the long island railroad is 57.5 inches.
The width of the railroad car on the long island railroad is 18 inches.
if two equations are the same at two different points, what can be said about the two equations?
Answer:
They are parallel
Step-by-step explanation:
They will have the same slope but never intersect
A man invested 1000 rupees,split onto two schemes,at annual rates of interest 8% and 9%.After one year he got 875 rupees as interests from both. How much did he invest in each?
Given :
Total amount, T = 10000 . ( It should not 1000 )
Annual rates of interest in both scheme is 8% and 9% .
After one year he got 875 rupees as interests from both.
To Find :
How much did he invest in each.
Solution :
Let, amount invested in 8% annual interest scheme is x.
So, amount in other scheme is 10000-x .
Now, sum of interest is equal to 875.
[tex]\dfrac{x\times r_1 \times 1}{100}+\dfrac{(10000-x)\times r_2\times 1}{100}=875\\\\8x + 9(10000-x)=87500\\\\x = 90000-87500\\\\x = 2500[/tex]
Therefore, amount invested in 8% and 9% annual interest scheme is 2500 and 7500 rupees respectively.
Hence, this is the required solution.
Is a quadrilateral a square?
Answer:
No but a square is a quadrilateral
Step-by-step explanation:
hope this helps :)
Answer:
If I were to be honest no Edit: yes it is
Step-by-step explanation:
No because a square has straight parrallel corners. But a quadrilateral has slanted parallel corners.
Hope that helped. But maybe wait for sumone else to answer too. : )
Edit: nvm my bad I was wrong
Someone for the love of god please help me.
Answer:
Step-by-step explanation:
(x + 10) cm
(x - 3) cm
The area of the rectangle is 48 cm².
Show that x satisfies the equation x2+ 7x - 78=0
Answer:
area a=lxb
48=(x+10)x(x-3)
48=x2-3x+10x-30
0=x2+7x-30-48
0=x2+7x-78
hence it is proved
The requried, we have shown that x satisfies the equation x² + 7x - 78 = 0.
What is the equation?The equation is the relationship between variables and represented as y = ax + b is an example of a polynomial equation.
Here,
The area of the rectangle is given by the formula:
Area = length x width
Substituting the given expressions for length and width, we have:
Area = (x + 10)(x - 3)
We are also told that the area of the rectangle is 48 cm², so we can set up the equation:
(x + 10)(x - 3) = 48
Expanding the left side using the FOIL method:
x² - 3x + 10x - 30 = 48
Simplifying:
x² + 7x - 78 = 0
Therefore, we have shown that x satisfies the equation x² + 7x - 78 = 0.
Learn more about equations here:
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Help me with this one
Answer:
SSS
Step-by-step explanation:
We can see that 2 sides are equal already, and because they share a side, that side is also equal. So, side side side.
100 points: Sketch a graph of y = 1x| Then sketch a graph of y = -x + 4 on the same coordinate plane. If you can change colors of your graph, make this one red. In the box below, describe the transformation that took place from y=[xl to y = -1x + 4|
Answer:
i cant like give you the answer off the top of my head but when i did this the video helped me a lot
Step-by-step explanation:
its How to Graph the line y = x by Cowan Academy
Solve the equation 186+10x=25.5x to find how many diving lessons
__ lesson(s)
Answer: x=12
Steps shown in image below:
What is the c.o.p. that describes the relationship between h and t.
The next section (This is the last five, don't worry xd)
7th grade math
I will give brainliest!
2 = a/6
find the value of a
Answer:
a = 12
Step-by-step explanation:
Given
2 = [tex]\frac{a}{6}[/tex] ( multiply both sides by 6 to clear the fraction )
12 = a
Answer:
a = 12
Step-by-step explanation:
We can do the inverse of division which is multiplication. 6 x 2 equals 12 and if we substitute 12 for a (12 divided by 6) you get 2 hope that helpss!!!