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:
Translate the sentence six less than the product of four and a number is -2 into an equation
Answer:
4x - 6 = -2
Step-by-step explanation:
i dont really know how to explain this one
Answer:
[tex]4x-6=-2[/tex]
Step-by-step explanation:
So since it says 6 is less than that means we will be putting the 6 in front of the number that's going to be 4x because the product of 4 and x is shown by 4x. Then lastly it says all "is -2" that means all that equals -2.
3/4 (4x+8)=-12. Distributive property
Answer:
x = -6
Step-by-step explanation:
I'm assuming you're solving for x
3/4 (4x + 8) = -12
you distribute the 3/4 to 4x and 8 so
3/4 times 4x = 3x
3/4 times 8 = 6
now looks like
3x + 6 = -12
3x = -12 - 6
3x = -18
x = -6
Which number best represents the slope of the graphed line?
Answer:
-1/3
Step-by-step explanation:
because i aint never seen two pretty best friends, its always one of em gotta be ugly
What is the mapping notation that will transform triangle ABC to triangle A'B'C'?
IF ANYBODY HAS ANSWERS TO THE TEST IM WILLING TO CASH APP
Answer:
Down 5 and left 3.
Step-by-step explanation:
Answer:
So it goes-8 for x and -5 for y aka to the left by 8 and down by 5
Step-by-step explanation:
Help please and please explain
Answer:
1st question example. all you have to do is write ∠XV, ∠VW, ∠WX
Step-by-step explanation:
the ∠ means the angles. so each angle has a degree. Just list them shortest angle to longest angle.
Express the radical using the imaginary unit, i.
Express your answer in simplified form.
IV-18 = +
Answer:
0.79
Step-by-step explanation:
25+30+10+4=79 so 0.79
Answer:
0.79
Step-by-step explanation:
I know because of I got taught a little about this but if I am wrong I'm sooooooo sorry
Is it possible for two triangles to have the same side lengths but not be congruent?
pls help me whats 45x57
Answer:
2565
Step-by-step explanation:
:)
Answer:
2565
Step-by-step explanation:
Hahshxnxjennshsne did. D d d s s d d d
Answer:
are you okay?
Step-by-step explanation:
is (3,2) a solution to 3x + 2y =13
Answer: yes
Step-by-step explanation:
(3*3) + (2*2) = 9+4 = 13 which is what the other side of the equation states so it is a solution
Answer:
Yes
Step-by-step explanation:
Plug the 3 into the x and plug the 2 into the y from (3,2)
So you get, 3(3) + 2(2)=13
9+4=13
plsss help me out i’m really having trouble a trying to figure this out
I'm pretty sure it's C. Both.
Answer:
only table A is a function
Table B has two inputes with the same outpute so it is impossible to be a function. but two of the same inputes can have different outputs
Deduce the square root of 7+4√3
2 -[tex]\sqrt{3}[/tex]
♥step by step♥7-4[tex]\sqrt{3}[/tex]
=7-2[tex]\sqrt{2x3}[/tex]
=7-2[tex]\sqrt{12}[/tex]
=([tex]\sqrt{3}[/tex])² - 2[tex]\sqrt{3x4}[/tex] + ([tex]\sqrt{4}[/tex] )
=( - [tex]\sqrt{3}[/tex] + [tex]\sqrt{4}[/tex])²
∴[tex]\sqrt{7-4}[/tex] [tex]\sqrt{3=2}[/tex] - [tex]\sqrt{3}[/tex]
♥plz give me brain list it would really help a lot! ♥
♥ お力になれて、嬉しいです！♥What is the product? [-3 4 2 -5]x[3 -2 1 0]
Answer:
71829x
Step-by-step explanation:
Answer:
The correct answer is option A
Step-by-step explanation
You got this! keep going! :D
I need helppppp I forgot how to do this :(
a=associates / p=partners
1a= $800 / 1p= $1500
(6)a + (2)p = 800 x 6 + 1500 + 2 = $7800
(x)a + (y)p = $z
hope this helps :)
Please help if you are good at math!!
Answer:
Y intercept= (0,15)
X intercept= ([tex]\frac{3}{4}[/tex],0) , (5,0)
Vertex= -[tex]\frac{b}{2a}[/tex]
Step-by-step explanation:
PLEASE MARK BRAINLIEST
He ran 10 meters in 2.7 seconds. priya ran 10 meters in 2.4 seconds.
who ran faster? explain how u know
Answer:
The first person somehow
Step-by-step explanation: 10 divide by 2.7 is 3.7037037 and 10 divided by 2.4 is 4.16666667
help me plz with algebra!!!
Answer:
Y= -12
Step-by-step explanation:
use the elimination method to solve the system of equations. choose the correct ordered pair x-y=2 2x+3y=14
Answer:
x = 4
y = 2
Step-by-step explanation:
x - y = 2
x = 2 + y
2x + 3y = 14
2(2 + y) + 3y = 14
4 + 2y + 3y = 14
4 + 5y = 14
5y = 10
y = 2
x - y = 2
x - 2 = 2
x = 4
Answer:
4,2
Step-by-step explanation:
Choose a justification for each step in the derivation of the sine difference identity.
Step 1:
Step 2: Distributive property
Step 3: Associative property
Step 4: Factoring out –1
Step 5:
Step 6:
Step 7:
The Sine difference function was derivated with the help of the sine sum function.
What is the sine difference function?The difference function for the sine function states that the sine of the difference of two angles equals the product of the sine of the first angle and cosine of the second angle minus the product of the cosine of the first angle and the sine of the second angle.
As we know that
[tex]Sin(a+b) = SinaCosb + CosaSinb[/tex]..........(1)
If we replace b by -b equation (1) becomes
[tex]Sin(a+(-b)) = SinaCos(-b) + CosaSin(-b)[/tex]
We know that
Cos(-b) =cosb
Sin(-b) =-Sinb
So, [tex]Sin(a-b) = SinaCosb -CosaSinb[/tex]
Therefore, the Sine difference function was derivated with the help of the sine sum function.
To get more about the Sine function visit:
https://brainly.com/question/10879727
Which equation is true when n = 1.2?
10n = 1.2
n + 1 = 1.2
5 + n = 6.2
1.2n = 10
Answer:
Step-by-step explanation:
ji
Answer:
5+n=6.2
Step-by-step explanation:
if n = 1.2
then its 5 +1+ .2 = 6.2 i think
slove the equation f-(-4)=12.
f =-8.
f =16.
f =8.
f = -9
Step-by-step explanation:
f-(-4)=12(Remove brackets)
f--4=12(Negative and negative is going to give you positive)
f+4=12(Group Like Terms)
f=12-4(Subtract)
f=8
Then that will be the final answer
Answer:
[tex]\huge\boxed{f=8}[/tex]
Step-by-step explanation:
[tex]f-(-4)=12\to f+4=12\qquad|(-)(-)\to(+)\\\\f+4=12\qquad|\text{subtract 4 from both sides}\\\\f+4-4=12-4\\\\f=8[/tex]
How many solutions does the system have?
y= -5x - 1
y = -5.2 + 7
Answer: y= -5x - 1
y = -5.2 + 7
Step-by-step explanation: y= -5x - 1 has 3. y = -5.2 + 7 has 1
Mrs. Kato has 6 bags of dried beans and a 2-pound bag of rice in one shopping bag. In another shopping bag she has a 5-pound bag of flour. The two shopping bags weigh the same amount. What is the weight of each bag of dried beans? Enter the correct answer in the box.
Given:
Mrs. Kato has 6 bags of dried beans and a 2-pound bag of rice in one shopping bag.
In another shopping bag she has a 5-pound bag of flour.
The two shopping bags weigh the same amount.
To find:
The weight of each bag of dried beans.
Solution:
Let x be the weight of each bag of dried beans.
So, weight of 6 bags of dried beans = 6x.
According to the question,
Weight of first bag = (6x + 2) pounds
Weight of second bag = 5 pounds
The two shopping bags weigh the same amount. So,
[tex]6x+2=5[/tex]
[tex]6x=5-2[/tex]
[tex]6x=3[/tex]
Divide both sides by 6.
[tex]x=\dfrac{3}{6}[/tex]
[tex]x=0.5[/tex]
Therefore, the weight of each bag of dried beans is 0.5 pounds.
The weight of each bag of dried beans is 0.5 pounds.
Given that,
Mrs. Kato has 6 bags of dried beans i.e. 6x and a 2-pound bag of rice in one shopping bag.In another shopping bag she has a 5-pound bag of flour. The two shopping bags weigh the same amount.Based on the above information, the calculation is as follows:
6x + 2 = 5
6x = 5 -2
6x = 3
x = 0.5
Therefore, we can conclude that the weight of each bag of dried beans is 0.5 pounds.
Learn more: brainly.com/question/24169758
I need help with this ASAP PLZ!!!!!!!!
Answer:
4. y = 2x -6
5. y = -x + 3
6. y = -[tex]\frac{1}{3}[/tex]x - 1
7. m= -4 and b=3
8. m= 1 and b=-7
9. y = 0.08x + 3
10. y= 3x - 19
11. y= -3x + 5
Step-by-step explanation:
Use the equation y = mx + b; where m is the slope and b is the y-intercept
4. given: m = 2 and y-int = -6; plug into equation
y = mx + b
y = 2x -6
5. given: m = -1 and y-int = 3; plug into equation
y = mx + b
y = -x + 3
6. given: m = -1/3 and y-int = -1; plug into equation
y = mx + b
y = -[tex]\frac{1}{3}[/tex]x - 1
7. given: y=-4x+3; slope m and y-intercept b are
m= -4
b=3
8. given: y=x-7; slope m and y-intercept b are
m= 1
b=-7
9. the monthly of $3 is fixed or starts the month; which is the y-intercept (another way to say is it is at 0 minutes) so b = 3. Now the rate is $0.08 per minutes and the rate is the same as the slope, m= 0.08
Using equation y=mx+b; substitute
y = 0.08x + 3
10. Given: (6,-1) and m= 3
use y=mx+b; substitute the y point into equation
-1 = 3x + b; substitue the x point into equation
-1 = 3(6) + b; solve for b
b = -19; now plug back into equation
y = mx + b or y= 3x - 19
11. Given: (2,-1) and m= -3
use y=mx+b; substitute the y point into equation
-1 = -3x + b; substitue the x point into equation
-1 = -3(2) + b; solve for b
b = 5; now plug back into equation
y = mx + b or y= -3x + 5
ILL GIVE BRAINLIEST + 3O POINTS
Estimate the value of each square root by finding a whole number greater than and a whole number less than the square root
√ 15
√ 24
√ 7
√ 48
√ 83
√ 105
Answer:
i guess you have to find square root of every number
like of 15 is 3,87 therefore higher than it is 4 and less than it 3
Answer:
4 and 35 and 43 and 27 and 610 and 911 and 10Step-By-Step Explanation: (Rounded to nearest hundred)
Square Root 15 = 3.87; rounded up: 4, rounded down: 3Square Root 24 = 4.90; rounded up: 5, rounded down: 4Square Root 7 = 2.65; rounded up: 3, rounded down : 2Square Root 48 = 6.93; rounded up: 7, rounded down: 6Square Root 83 = 9.10; rounded up: 10, rounded down: 9Square Root 105 = 10.24; rounded up: 11, rounded down: 10i will give 100 pionts and branlyest to anyone who gits this right
If a scale drawing uses a scale of 1:12, then 2 1/2 inches on the model represents 2 1/2 feet. TrueFalse
Answer:
true
Step-by-step explanation:
Solve using Elimination
3x+y=-9
y=5x+7
Answer
x=-2 and y=-3
Step-by-step explanation:
Step: Solvey=5x+7for y:
Step: Substitute5x+7 for y in 3x+y=−9:
3x+y=−9
3x+5x+7=−9
8x+7=−9(Simplify both sides of the equation)
8x+7+−7=−9+−7(Add -7 to both sides)
8x=−16
8x/8=−16/8
(Divide both sides by 8)
x=−2
Step: Substitute−2 for x in y=5x+7:
y=5x+7
y=(5)(−2)+7
y=−3(Simplify both sides of the equation)
Solve please
18 + 8x<34
Answer:
x=2
Step-by-step explanation:
Which system of equations represents the graph?
Answer:
B
Step-by-step explanation:
if you look at the line one going up, the y axis is at the negatives meaning it begins as -5 . Then notice how it goes up 3 times and it goes to the right once. So the slope is 3/1 or simplified 3 so we know the slope intercept form is = y is 3x-5. Now we look at the second one. None of the answers of the second one are in slope intercept form. for that you have to turn them into it. When you do that for B will be y= -1/2x + 2. When you look at the line the y axis is 2 and the slope is -1/2x.
How would you write 300000 + 400000 + 600 + 7 in standard form?
Answer:
700607
Step-by-step explanation: