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4 years ago

6 times a number subtracted from 49 is 5

Mathematics

2 answers:

vitfil [10]4 years ago

8 0

The answer is 9 because 6 times 9 equals 54 and then subtracted from 49 equals 5

*Hope I Helped*

Dmitry [639]4 years ago

4 0

49-6n=5 subtract 49 from both sides -6n= -44 divide by -6 n= 22/3 hope this helps

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show that the curve has 3 points of inflection and they all lie on 1 straight line:\[y=\frac{1+x}{1+x^{2}}\]

ohaa [14]

Y = (1 + x) / (1 + x^2)

y'
= [(1 + x^2)(1) - (1 + x)(2x)] / (1 + x^2)^2
= [1 + x^2 - 2x - 2x^2] / (1 + x^2)^2
= [-x^2 - 2x + 1] / (1 + x^2)^2

y''
= [(1 + x^2)^2 * (-2x - 2) - (-x^2 - 2x + 1)(2)(1 + x^2)(2x)] / (1 + x^2)^4
= [(1 + x^2)(-2x - 2) - (4x)(-x^2 - 2x + 1)] / (1 + x^2)^3
= [(-2x - 2x^3 - 2 - 2x^2) - (-4x^3 - 8x^2 + 4x)] / (1 + x^2)^3
= [-2x - 2x^3 - 2 - 2x^2 + 4x^3 + 8x^2 - 4x] / (1 + x^2)^3
= [2x^3 + 6x^2 - 6x - 2] / (1 + x^2)^3

Setting y'' to zero, we have:
y'' = 0
[2x^3 + 6x^2 - 6x - 2] / (1 + x^2)^3 = 0
(2x^3 + 6x^2 - 6x - 2) = 0

Using trial and error, you will realise that x = 1 is a root.
This means (x - 1) is a factor.
Dividing 2x^3 + 6x^2 - 6x - 2 by x - 1 using long division, you will have 2x^2 + 8x + 2.

2x^2 + 8x + 2
= 2(x^2 + 4x) + 2
= 2(x + 2)^2 - 2(2^2) + 2
= 2(x + 2)^2 - 8 + 2
= 2(x + 2)^2 - 6

Setting 2x^2 + 8x + 2 to zero, we have:
2(x + 2)^2 - 6 = 0
2(x + 2)^2 = 6
(x + 2)^2 = 3
x + 2 = sqrt(3) or = -sqrt(3)
x = -2 + sqrt(3) or x = -2 - sqrt(3)

Note that -2 - sqrt(3) < -2 + sqrt(3) < 1
We will choose random values belonging to each interval and test them out.

-5 < -2 - sqrt(3) < -2 < -2 + sqrt(3)
f''(-5) = [2(-5)^3 + 6(-5)^2 - 6(-5) - 2] / (1 + (-5)^2)^3 = -9/2197 < 0
f''(-2) = [2(-2)^3 + 6(-2)^2 - 6(-2) - 2] / (1 + (-2)^2)^3 = 18/125 > 0
Note that one value is positive and the other is negative.
Thus, x = -2 - sqrt(3) is an inflection point.

-2 - sqrt(3) < -2 < -2 + sqrt(3) < 0 < 1
f''(-2) = [2(-2)^3 + 6(-2)^2 - 6(-2) - 2] / (1 + (-2)^2)^3 = 18/125 > 0
f''(0) = [2(0)^3 + 6(0)^2 - 6(0) - 2] / (1 + (0)^2)^3 = -2 < 0
Note that one value is positive and the other is negative.
Thus, x = -2 + sqrt(3) is also an inflection point.

-2 + sqrt(3) < 0 < 1 < 2
f''(0) = [2(0)^3 + 6(0)^2 - 6(0) - 2] / (1 + (0)^2)^3 = -2 < 0
f''(2) = [2(2)^3 + 6(2)^2 - 6(2) - 2] / (1 + (2)^2)^3 = 26/125 > 0
Note that one value is positive and the other is negative.
Thus, x = 1 is an inflection point.

Hence, we have three inflection points in total.

When x = -2 - sqrt(3), we have:
y
= (1 - 2 - sqrt(3)) / (1 + (-2 - sqrt(3))^2)
= (-1 - sqrt(3)) / (1 + 4 + 4sqrt(3) + 3)
= (-1 - sqrt(3)) / (8 + 4sqrt(3))

When x = -2 + sqrt(3), we have:
y
= (1 - 2 + sqrt(3)) / (1 + (-2 + sqrt(3))^2)
= (-1 + sqrt(3)) / (1 + 4 - 4sqrt(3) + 3)
= (-1 + sqrt(3)) / (8 - 4sqrt(3))

When x = 1, we have:
y
= (1 + 1) / (1 + 1^2)
= 2 / 2
= 1

Using the slope formula, we have:
(y - 1) / (x - 1) = [[(-1 + sqrt(3)) / (8 - 4sqrt(3))] - 1] / ( -2 + sqrt(3) - 1)
(y - 1) / (x - 1) = 1/4, which is the equation of the line which the inflection points at x = 1 and x = -2 + sqrt(3) lies on.

Note that I am skipping the intermediate steps for simplifying here, but the trick is to rationalise the denominator by multiplying a conjugate on both numerator and denominator.

Now, we just need to check that the inflection point at x = -2 - sqrt(3) lies on the same line as well.
L.H.S.
= [[(-1 - sqrt(3)) / (8 + 4sqrt(3))] - 1] / (-2 - sqrt(3) - 1)
= 1/4
= R.H.S.

Once again, I am skipping simplifying steps here.

Anyway, this proves all three points of inflection lies on the same straight line.

4 0

3 years ago

NEED ANSWER ASAP PLEASE!!

erastova [34]

Answer:

DE=5*8=40

Step-by-step explanation:

DE+EF=DF

5x+17x-20=20x-4

22x-20=20x-4

22x-20x=-4+20

2x=16

x=16/2

x=8

DE=5x

DE=5*8=40

6 0

3 years ago

The slope of the line below is 2. Which of the following is the point-slope form of the line?

Allisa [31]

Point-slope form is y1 - y2 = m(x1 - x2). We know that the slope is 2, and we have a point of (1, -1). If we plug in our numbers, we will get y -(-1) = 2(x - 1). This gives us y + 1 = 2(x - 1). Therefore, the answer is B.

4 0

3 years ago

Read 2 more answers

Write a algebraic expression

iren [92.7K]

Answer:

1. $a^{3}b^{4}$ .

2. $3Y^{2} - 6$ .

3. $a + b + \frac{b}{a}$ .

4. $4(r + a) + 2(r - s)$ .

5. $3(55 - w^{3})$ .

Step-by-step explanation:

1. The product of the third power of a and the fourth power of b = $a^{3}b^{4}$ .

The 'third power of a' means 'a to the power 3' and 'fourth power of b' means 'b to the power 4'.

2. Six less than three times the square of Y means $3Y^{2} - 6$ .

Here, 'three times the square of Y' means 'product of 3 and Y to the power 2', and 'six less than' means 'subtract 6 from the expression'.

3. The sum of a and b increase by the quotient of b and a means $a + b + \frac{b}{a}$ .

Here, 'quotient of b and a' means 'b divided by a'.

4. Four times the sum of r and a increased by twice the different r and s means $4(r + a) + 2(r - s)$ .

Here, 'increased by' means 'addition' and 'different r and s' means 'r minus s'.

5. Triple the difference of 55 and cube of w means $3(55 - w^{3})$ .

Here, triple means 3 times. (Answer)

7 0

4 years ago

PLEASE ANSWER

statuscvo [17]

Answer: B) 2

Detailed Answer:

The degree of a polynomial is the highest power in the polynomial. Here the highest power is 2 hence the degree is 2. Polynomials with the degree 2 are also called quadratic polynomials.

8 0

2 years ago