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Natalka [10]
3 years ago
13

Consider the polynomial p(s) = s2 + bs + c where b and c are real numbers. Show that all the roots of p(s) are both contained in

the open left half plane {s : s < 0} if and only if b > 0 and c > 0. Hint: use the quadratic formula.
Mathematics
2 answers:
hodyreva [135]3 years ago
7 0

Answer:

It is shown in the explanation

Step-by-step explanation:

p(s) = s² + bs + c

a = 1

b = b

c = c

We get Δ as follows

Δ = (b²-4*a*c) = b² - 4*1*c = b² - 4c > 0  ⇒   b² > 4c  ⇔ c > 0

s = (-b + √(b² - 4c))/2(1)

⇒   s₁ = (-b + √(b² - 4c))/2

s₂ = (-b - √(b² - 4c))/2(1)

⇒   s₂ = (-b - √(b² - 4c))/2

We have that -b < 0  ⇔ b > 0

then s₁ < 0 and s₂ < 0 ⇔ c > 0 and b > 0

Taya2010 [7]3 years ago
6 0

Answer:

For roots to lie on the left half plane, b ⊃ 0 and c ⊃0

Step-by-step explanation:

From quadratic formula, we have;

x = -b±√(b²-4ac)/2a

From the given expression p(s) = s² + bs + c,

x = s

a = 1

b = b

c = c

The quadratic formula can then be written as;

s =  -b±√(b²-4*1*c)/2*1

   =  -b±√(b²-4c)/2

s₁ =  -b+√(b²-4c)/2

s₂ =  -b±√(b²-4c)/2

From the equation above,

Sum of root = -b

Product of root = c

If both the root lie on left side of the s-plane, then sum of roots will be negative. Hence, -b ∠0. That is, b ⊃0

Also, the product root will be positive, c ⊃ 0

Hence, for roots to lie on the left half plane, b ⊃ 0 and c ⊃0

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Evgesh-ka [11]

The equation of the circle with center at (2,-4) and tangent to the line x = -2 is given by:

(2) (x - 2)² + (y + 4)² = 16

<h3>What is the equation of a circle?</h3>

The equation of a circle of center (x_0, y_0) and radius r is given by:

(x - x_0)^2 + (y - y_0)^2 = r^2

The circle has center at (2,-4), hence x_0 = 2, y_0 = -4.

The center is tangent to the line x = -2, hence it has a point at x = -2, since |-2 - 2| = 4, r = 4 -> r² = 16.

Hence the equation of the circle is given by:

(2) (x - 2)² + (y + 4)² = 16

More can be learned about the equation of a circle at brainly.com/question/24307696

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6 0
2 years ago
Help is needed please help
Anna007 [38]

You have to do Pythagorean theorem:

a^{2} +b^{2}  = c^{2}

"a" and "b" are the legs. In this case 7 is a leg and x is a leg

"c" is the hypotenuses, which is 19 here

so...

a = 7

b = x

c = 19

Plug the numbers above into the equation:

7^{2} + x^{2} = 19^{2}

49 +x^{2} = 361

Isolate x by subtracting 49 to both sides:

(49-49) + x^{2} = 361 - 49

x^{2} = 312

To solve for x you have to take the square root to both sides of the equation:

\sqrt{x^{2} } = \sqrt{312}

x = 17.6635

x = 17.7

Hope this helped!

6 0
3 years ago
4x-5=2x-3
Lemur [1.5K]
X = 4 Hope You In Joy I Did The Math My Self
7 0
3 years ago
This year, jake is 5 years older than his sister. Three years ago, jake was twice his sisters age. How old is jakes sister now?
ruslelena [56]

you don't have enough info, we need to know how old he was when he was twice her age, or how old she was

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3 years ago
What is the value of X
Volgvan

Answer:

Step-by-step explanation:

it's another one of those similar triangles

it's easier to see this one.

t /c = u / b

where

t = 7x + 3

c = 27

u = 35

b = 21

plug in the knowns

7x + 3 / 27 = 35 / 21

7x + 3  = 27 ( 35 / 21 )

7x + 3 = 9 ( 35 / 7 )     ( divided by 3 )

7x  + 3  = 9 ( 5)     ( 7 into 35 =5 )

7x  + 3  = 45

7x = 45 - 3

7x = 42  

x = 6

I checked it, the ratio is 1_2/3  :) for both, but you can check it too

7 0
3 years ago
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