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Leya [2.2K]
3 years ago
14

Prove that the roots of x2+(1-k)x+k-3=0 are real for all real values of k​

Mathematics
1 answer:
masha68 [24]3 years ago
6 0

Answer:

Roots are not real

Step-by-step explanation:

To prove : The roots of x^2 +(1-k)x+k-3=0x

2

+(1−k)x+k−3=0 are real for all real values of k ?

Solution :

The roots are real when discriminant is greater than equal to zero.

i.e. b^2-4ac\geq 0b

2

−4ac≥0

The quadratic equation x^2 +(1-k)x+k-3=0x

2

+(1−k)x+k−3=0

Here, a=1, b=1-k and c=k-3

Substitute the values,

We find the discriminant,

D=(1-k)^2-4(1)(k-3)D=(1−k)

2

−4(1)(k−3)

D=1+k^2-2k-4k+12D=1+k

2

−2k−4k+12

D=k^2-6k+13D=k

2

−6k+13

D=(k-(3+2i))(k+(3+2i))D=(k−(3+2i))(k+(3+2i))

For roots to be real, D ≥ 0

But the roots are imaginary therefore the roots of the given equation are not real for any value of k.

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You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. Th
Goshia [24]

The probability that this bag will be warm enough on a randomly

selected May night at the park is 0.8106 ⇒ answer C

Step-by-step explanation:

You are planning a May camping trip to Denali National Park in Alaska

and want to make sure your sleeping bag is warm enough.

The average low temperature in the park for May follows a normal

distribution

The given is:

1. The mean is 32°F

2. The standard deviation is 8°F

3. One sleeping bag you are considering advertises that it is good for

   temperatures down to 25°F (x ≥ 25)

At first let us find the z-score

∵ z = (x - μ)/σ, where x is the score, μ is the mean, σ is standard deviation

∵ μ = 32°F , σ = 8°F , x = 25°F

- Substitute these values in the rule

∴ z = \frac{25-32}{8}=-0.875

Now let us find the corresponding area of z-score in the normal

distribution table

∵ The corresponding area of z = -0.875 is 0.18943

∵ For P(x ≥ 25) the area to the right is needed

∵ P(x ≥ 25) = 1 - 0.18943 = 0.8106

∴ P(x ≥ 25) = 0.8106

The probability that this bag will be warm enough on a randomly

selected May night at the park is 0.8106

Learn more:

You can learn more about mean and standard deviation in brainly.com/question/6073431

#LearnwithBrainly

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