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SashulF [63]
3 years ago
9

Find the range of p in equation p(x+1) (x-3)=x-4p-2 has no real roots. ​

Mathematics
1 answer:
Tanya [424]3 years ago
4 0

Answer:

\displaystyle p > \frac{1}{4}.

Step-by-step explanation:

Expand the left-hand side of this equation:

p\, (x + 1)\, (x - 3) = p\, \left(x^2 - 2\, x - 3\right) = p\, x^2 - 2\, p\, x - 3\, p.

Collect the terms, so that this quadratic equation is in the form a\, x^2 + b\, x+ c = 0:

p\, x^2 - 2\, p\, x - 3\, p = x - 4\, p - 2.

p\, x^2 - (2\, p + 1)\, x + (p + 2) = 0.

In this equation:

  • a = p.
  • b = -(2\, p + 1).
  • c = p + 2.

Calculate the quadratic discriminant of this quadratic equation:

\begin{aligned}b^2 - 4\, a\, c &= (-(2\, p + 1))^2 - 4\, p\, (p + 2) \\ &= 4\, p^2 + 4\, p + 1 - 4\, p^2- 8\, p = -4\, p + 1\end{aligned}.

A quadratic equation has no real root if its quadratic discriminant is less then zero. As a result, this quadratic equation will have no real root when -4\, p + 1 < 0. Solve for the range of p:

\displaystyle p > \frac{1}{4}.

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Bottles of mango juice are assumed to contain 275 milliliters of juice. There is some variation from bottle to bottle because th
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Answer:

Step-by-step explanation:

The mean of the set of data given is

Mean = (275.4 + 276.8 + 273.9 + 275.0 + 275.8 + 275.9 + 276.1)/7 = 275.56

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Summation(x - mean)^2 = (275.4 - 275.56)^2 + (276.8 - 275.56)^2 + (273.9 - 275.56)^2 + (275.0 - 275.56)^2 + (275.8 - 275.56)^2 + (275.9 - 275.56)^2 + (276.1 - 275.56)^2 = 5.0972

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We would set up the hypothesis test.

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This is a 2 tailed test.

Since the number of samples is small and no population standard deviation is given, the distribution is a student's t.

Since n = 7

Degrees of freedom, df = n - 1 = 7 - 1 = 6

t = (x - µ)/(s/√n)

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x = sample mean = 275.56

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s = samples standard deviation = 0.85

t = (275.56 - 275)/(0.85/√7) = 1.74

We would determine the p value using the t test calculator. It becomes

p = 0.132

Because the p-value of 0.132 is greater than the significance level of 0.05, we would fail to reject the null hypothesis. We conclude the data does not provide convincing evidence that the mean amount of juice in all the bottles filled that day differs from the target value of 275 milliliters.

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