If -1 is subtracted from 5, what will be the result?

What is the probability that all the roots of x2 + bx + c = 0 are real? [0,1]?

notsponge [240]

$x^2+bx+c$ will have real roots when the discriminant of the quadratic, $\Delta=b^2-4c$ , is non-negative, i.e.

$b^2-4c\ge0\implies b^2\ge4c$

Your question about probability is currently impossible to answer without knowing exactly what the experiment is. Are you picking $b,c$ at random from some interval? Is the choice of either distributed a certain way?

I'll assume the inclusion of "[0,1]" in your question is a suggestion that both $b,c$ are chosen indepently of one another from [0, 1]. Let $B,C$ denote the random variables that take on the values of $b,c$ , respectively. I'll assume $B,C$ are identical and follow the standard uniform distribution, i.e. they each have the same PDF and CDF as below:

$f_X(x)=\begin{cases}1&\text{for }0$

$F_X(x)=\begin{cases}0&\text{for }x$

where $X$ is either of $B,C$ .

Then the question is to find $P(B^2\ge4C)$ . We have

$P(B^2\ge4C)=P\left(C\le\dfrac{B^2}4\right)$

and we can condition the random variable $C$ on the event of $B=b$ by supposing

$P\left(C\le\dfrac{B^2}4\right)=P\left(\left(C\le\dfrac{B^2}4\right)\land(B=b)\right)=P\left(C\le\dfrac{B^2}4\mid B=b\right)\cdot P(B=b)$

then integrate over all possible values of $b$ .

$=\displaystyle\int_{-\infty}^\infty P\left(C\le\dfrac{b^2}4\right)f_B(b)\,\mathrm db$

$=\displaystyle\int_{-\infty}^\infty F_C\left(\dfrac{b^2}4\right)f_B(b)\,\mathrm db$

$=\displaystyle\int_0^1\dfrac{b^2}4\,\mathrm db=\frac1{12}$

4 0

3 years ago

Select the correct answer. Given the following formula, solve for a. A. B. C. D. s=a+b+C/2

nexus9112 [7]

Hey there!

The answer to your question is a = s - b - C/2

To solve for "a", we must get "a" on it's own side of the equal sign. We can first subtract "s" from both sides:

0 = a + b + C/2 - s

Then, we can subtract a from both sides:

-a = b + C/2 - s

Finally, we divide both sides by -1, so a is by itself.

a = -b - C/2 + s

And we can rearrange it to the standard form:

a = s - b - C/2

Good luck, and hope it helped! Have a great day!

7 0

3 years ago

Which shows a perfect square trinomial?

astra-53 [7]

Answer:

100-36x^2y^2

Step-by-step explanation:

100-36x^2y^2 =

(10-6xy)(10+6xy)

Because x^2-y^2 = (x+y)(x-y)

7 0

4 years ago

Read 2 more answers

Which of the following is true of a radius of a circle?

Sav [38]

The second one is correct (Radius is the distance from the center to any point on the circle)

6 0

3 years ago

Read 2 more answers

Find m (-3x + 20) (-2x + 55)

Sergeeva-Olga [200]

Answer: Solution for (-3x+20)+(-2x+55)=180 equation: x= -21

Step-by-step explanation:

Simplifying

(-3x + 20) + (-2x + 55) = 180

Reorder the terms:

(20 + -3x) + (-2x + 55) = 180

Remove parenthesis around (20 + -3x)

20 + -3x + (-2x + 55) = 180

Reorder the terms:

20 + -3x + (55 + -2x) = 180

Remove parenthesis around (55 + -2x)

20 + -3x + 55 + -2x = 180

Reorder the terms:

20 + 55 + -3x + -2x = 180

Combine like terms: 20 + 55 = 75

75 + -3x + -2x = 180

Combine like terms: -3x + -2x = -5x

75 + -5x = 180

Solving

75 + -5x = 180

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-75' to each side of the equation.

75 + -75 + -5x = 180 + -75

Combine like terms: 75 + -75 = 0

0 + -5x = 180 + -75

-5x = 180 + -75

Combine like terms: 180 + -75 = 105

-5x = 105

Divide each side by '-5'.

x = -21

Simplifying

x = -21

6 0

3 years ago