Easy question about multiplication? unsure why not getting full marks

Answer:

The idea is to transform the expression by multiplying [tex](\sqrt{x + 1} - \sqrt{x})[/tex] with its conjugate, [tex](\sqrt{x + 1} + \sqrt{x})[/tex].

Step-by-step explanation:

For any real number [tex]a[/tex] and [tex]b[/tex], [tex](a + b)\, (a - b) = a^{2} - b^{2}[/tex].

The factor [tex](\sqrt{x + 1} - \sqrt{x})[/tex] is irrational. However, when multiplied with its square root conjugate [tex](\sqrt{x + 1} + \sqrt{x})[/tex], the product would become rational:

[tex]\begin{aligned} & (\sqrt{x + 1} - \sqrt{x}) \, (\sqrt{x + 1} + \sqrt{x}) \\ &= (\sqrt{x + 1})^{2} -(\sqrt{x})^{2} \\ &= (x + 1) - (x) = 1\end{aligned}[/tex].

The idea is to multiply [tex]\sqrt{x}\, (\sqrt{x + 1} - \sqrt{x})[/tex] by [tex]\displaystyle \frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}[/tex] so as to make it easier to take the limit.

Since [tex]\displaystyle \frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}} = 1[/tex], multiplying the expression by this fraction would not change the value of the original expression.

[tex]\begin{aligned} & \lim\limits_{x \to \infty} \sqrt{x} \, (\sqrt{x + 1} - \sqrt{x}) \\ &= \lim\limits_{x \to \infty} \left[\sqrt{x} \, (\sqrt{x + 1} - \sqrt{x})\cdot \frac{\sqrt{x + 1} + \sqrt{x}}{\sqrt{x + 1} + \sqrt{x}}\right] \\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x}\, ((x + 1) - x)}{\sqrt{x + 1} + \sqrt{x}} \\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x + 1}+ \sqrt{x}}\end{aligned}[/tex].

The order of [tex]x[/tex] in both the numerator and the denominator are now both [tex](1/2)[/tex]. Hence, dividing both the numerator and the denominator by [tex]x^{(1/2)}[/tex] (same as [tex]\sqrt{x}[/tex]) would ensure that all but the constant terms would approach [tex]0[/tex] under this limit:

[tex]\begin{aligned} & \lim\limits_{x \to \infty} \sqrt{x} \, (\sqrt{x + 1} - \sqrt{x}) \\ &= \cdots\\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x + 1}+ \sqrt{x}} \\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x} / \sqrt{x}}{(\sqrt{x + 1} / \sqrt{x}) + (\sqrt{x} / \sqrt{x})} \\ &= \lim\limits_{x \to \infty}\frac{1}{\sqrt{(x / x) + (1 / x)} + 1} \\ &= \lim\limits_{x \to \infty} \frac{1}{\sqrt{1 + (1/x)} + 1}\end{aligned}[/tex].

By continuity:

[tex]\begin{aligned} & \lim\limits_{x \to \infty} \sqrt{x} \, (\sqrt{x + 1} - \sqrt{x}) \\ &= \cdots\\ &= \lim\limits_{x \to \infty} \frac{\sqrt{x}}{\sqrt{x + 1}+ \sqrt{x}} \\ &= \cdots \\ &= \lim\limits_{x \to \infty} \frac{1}{\sqrt{1 + (1/x)} + 1} \\ &= \frac{1}{\sqrt{1 + \lim\limits_{x \to \infty}(1/x)} + 1} \\ &= \frac{1}{1 + 1} \\ &= \frac{1}{2}\end{aligned}[/tex].

Answer:

Hello,

Step-by-step explanation:

[tex]\displaystyle \lim_{x \to \infty} \sqrt{x}*(\sqrt{x+1}-\sqrt{x} ) \\\\\\= \lim_{x \to \infty}\dfrac{ \sqrt{x}*(\sqrt{x+1}-\sqrt{x} )*(\sqrt{x+1}+\sqrt{x} )}{\sqrt{x+1} +\sqrt{x} } \\\\= \lim_{x \to \infty} \dfrac{\sqrt{x} *1}{\sqrt{x+1} +\sqrt{x} } \\\\\\= \lim_{x \to \infty} \dfrac{1} {\sqrt {\dfrac {x+1} {x} }+\sqrt{\dfrac{x}{x} } } \\\\\\=\dfrac{1} {\sqrt {1}+\sqrt{1} } \\\\\\=\dfrac{1} {2} \\[/tex]

Answer:

D. -4x(x - 2)(x+3)

Step-by-step explanation:

We are given the following trinomial:

[tex]-4x^3 - 4x^2 + 24x[/tex]

-4x is the common term, so:

[tex]-4x(\frac{-4x^3}{-4x} - \frac{4x^2}{-4x^3} + \frac{24x}{-4x}) = -4x(x^2+x-6)[/tex]

The second degree polynomial can also be factored, finding it's roots.

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

[tex]ax^{2} + bx + c, a\neq0[/tex].

This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = a(x - x_{1})*(x - x_{2})[/tex], given by the following formulas:

[tex]x_{1} = \frac{-b + \sqrt{\Delta}}{2*a}[/tex]

[tex]x_{2} = \frac{-b - \sqrt{\Delta}}{2*a}[/tex]

[tex]\Delta = b^{2} - 4ac[/tex]

x² + x - 6

Quadratic equation with [tex]a = 1, b = 1, c = -6[/tex]

So

[tex]\Delta = 1^{2} - 4(1)(-6) = 25[/tex]

[tex]x_{1} = \frac{-1 + \sqrt{25}}{2} = 2[/tex]

[tex]x_{2} = \frac{-1 - \sqrt{25}}{2} = -3[/tex]

So

[tex]x^2 + x - 6 = (x - 2)(x - (-3)) = (x - 2)(x + 3)[/tex]

The complete factorization is:

[tex]-4x(x^2+x-6) = -4x(x - 2)(x + 3)[/tex]

Thus the correct answer is given by option d.

Answer:

A. x = 3

Step-by-step explanation:

5x - 3 = 12

5x = 12 + 3

5x = 15

x = 15/5

= 3

9514 1404 393

Answer:

  -1/2

Step-by-step explanation:

The factor outside parentheses multiplies each term inside.

  5/2(-8/5 +7/5)

  = (5/2)(-8/5) +(5/2)(7/5)

  = -8/2 +7/2 = -1/2

Answer:

6x10^4

Step-by-step explanation:

Answer:

The correct graph is A.

Answer:

A i got it right

Step-by-step explanation:

Answer:

x=27

Step-by-step explanation:

The sum of the angles of a triangle are 180 degrees

90 + x+15 + 2x-6 = 180

Combine like terms

3x+99=180

Subtract 99 from each side

3x+99-99=180-99

3x =81

Divide each side by 3

3x/3 = 81/3

x=27

Here we have the quadratic function:

f(x) = -16*x^2 + 60*x + 16

We can see that it is in standard form:

y = a*x^2 + b*x + c

a) First we want to completely factorize the function f(x).

To do it, we first need to find the roots of f(x).

Remember that for a generic quadratic equation:

a*x^2 + b*x + c = 0

whit roots x₁ and x₂, the factorized form is:

a*(x - x₁)*(x - x₂)

And the roots are given by:

[tex]x = \frac{-b \pm \sqrt{b^2 - 4*a*c} }{2*a}[/tex]

Then for the case of f(x) = -16*x^2 + 60*x + 16, the roots are:

[tex]x = \frac{-60 \pm \sqrt{60^2 - 4*(-16)*16} }{2*(-16)} = \frac{-60 \pm 68}{-32}[/tex]

So the two roots are:

x₁ = (-60 + 68)/-32 = -0.25

x₂ = (-60 - 68)/-32 = 4

Then the factorized form is:

f(x) = -16*(x - 4)*(x + 0.25)

B) We already found the roots, which are:

x₁ =  -0.25

x₂ =  4

These are the x-intercepts:

(-0.25, 0) and (4, 0)

C) We can see that the leading coefficient is negative.

This means that the arms of the graph go downwards, so as |x| increases, the value of f(x) tends to negative infinity.

D) To graph f(x) we can find some of the points of the graph (like the x-intercepts and some more of them) and then connect them with a parabola curve, the graph that you will get is the one that you can see below.

If you want to learn more about this topic, you can read:

https://brainly.com/question/22761001

Answer:

67.98,68.11, 68.6

Answer: x=52.6°

Step-by-step explanation:

To find the value of x, we have to use our SOHCAHTOA. We can eliminate sine and cosine because both uses hypotenuse, which is not labelled. Therefore, we use tangent.

[tex]tan(x)=\frac{17}{13}[/tex]

To find x, we want to use inverse tangent.

[tex]x=tan^{-1}(\frac{17}{13} )[/tex]      [plug into calculator]

[tex]x=52.6[/tex]

Now, we know that x=52.6°.

Answer:

A the input x=3 goes to two different output values

Step-by-step explanation:

This is not a function

x = 3 goes to two different y values

x = 3 goes to t = 10 and y = 5

Answer:

3rd option (top right)

Step-by-step explanation:

3rd option represents a linear equation

y = -2x-1

Answered by GAUTHMATH

Answer:

Carlos

Step-by-step explanation:

Hope this helps

Answer:

the correct answer is, 4

Answer:

False

Step-by-step explanation:

Before testing the hypothesis, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

West University:

15 out of 100, so:

[tex]p_W = \frac{15}{100} = 0.15[/tex]

[tex]s_W = \sqrt{\frac{0.15*0.85}{100}} = 0.0357[/tex]

East University:

12 out of 100, so:

[tex]p_E = \frac{12}{100} = 0.12[/tex]

[tex]s_E = \sqrt{\frac{0.12*0.88}{100}} = 0.0325[/tex]

Test the difference in driving abilities at the two universities:

At the null hypothesis we test if there is no difference, that is, the subtraction of the proportions is 0, so:

[tex]H_0: p_W - p_E = 0[/tex]

At the alternative hypothesis, we test if there is a difference, that is, if the subtraction of the proportions is different of 0. So

[tex]H_1: p_W - p_E \neq 0[/tex]

The test statistic is:

[tex]z = \frac{X - \mu}{s}[/tex]

In which X is the sample mean, [tex]\mu[/tex] is the value tested at the null hypothesis, and s is the standard error.

0 is tested at the null hypothesis:

This means that [tex]\mu = 0[/tex]

From the two samples:

[tex]X = p_W - p_E = 0.15 - 0.12 = 0.03[/tex]

[tex]s = \sqrt{s_W^2+s_E^2} = \sqrt{0.0357^2+0.0325^2} = 0.0483[/tex]

Value of the test statistic:

[tex]z = \frac{X - \mu}{s}[/tex]

[tex]z = \frac{0.03 - 0}{0.0483}[/tex]

[tex]z = 0.62[/tex]

P-value of the test and decision:

The p-value of the test is the probability that the proportions differ by at least 0.03, which is P(|z| > 0.62), that is, 2 multiplied by the p-value of z = -0.62.

Looking at the z-table, z = -0.62 has a p-value of 0.2676.

2*0.2676 = 0.5352.

The p-value of the test is 0.5352 > 0.05, which means that the difference in driving is not statistically significant at the .05 significance level, and thus the answer is False.

Answer:

8 sqrt(5)

Step-by-step explanation:

sqrt(45) + sqrt(125)

Rewriting

sqrt(9*5) + sqrt( 25 *5)

we know sqrt(ab) = sqrt(a) sqrt(b)

sqrt(9) sqrt(5) + sqrt(25) sqrt(5)

3 sqrt(5)+5 sqrt(5)

Add like terms

8 sqrt(5)

Answer:

A.    [tex] 8\sqrt{5} [/tex]

Step-by-step explanation:

[tex] \sqrt{45} + \sqrt{125} = [/tex]

[tex] = \sqrt{9 \times 5} + \sqrt{25 \times 5} [/tex]

[tex] = 3\sqrt{5} + 5\sqrt{5} [/tex]

[tex] = 8\sqrt{5} [/tex]

Answer:

$48

Explanation:

> 50 x .20 = $10

$50 + $10= $60

-----------------------------

> 60 x .20 = $12

$60 - $12= $48

Answer: A

Step-by-step explanation:

(tangent is opposite over adjacent)

[tex]tan(40)=\frac{x}{3.8}\\x=3.8*tan(40)[/tex]

9514 1404 393

Answer:

  false

Step-by-step explanation:

The statement is nonsense (false). Regardless of the sign of a product, subtraction plays no part in anything related to it.

Answer:

3/85

Step-by-step explanation:

that's the answer above

Answer:

55°

Step-by-step explanation:

Vertical angles are similar

Answered by GAUTHMATH

9514 1404 393

Answer:

Step-by-step explanation:

Reflection over the line x=a is the transformation ...

  (x, y) ⇒ (2a -x, y)

Then the double reflection over x=a and x=b is the transformation ...

  (x, y) ⇒ (2b -(2a -x), y) = (2(b-a) +x, y)

That is, the result is translation by twice the distance between the lines. For a=1 and b=3, the transformation is ...

  (x, y) ⇒ (x +4, y) . . . . . . . translation to the right by 4 units.

  A(-5, 2) ⇒ A''(-1, 2)

  B(-1, 5) ⇒ B''(3, 5)

  C(0, 3) ⇒ C''(4, 3)

Answer:

Step-by-step explanation:

Answer:

The answer is "1900"

Step-by-step explanation:

It takes 500 fewer calories per day for her to lose 1 lb of weight every week.

[tex]\to (15 \times 160)-500 =(2400)-500 =2400-500=1900[/tex]

Answer:

Point T,R,P not seen how I don't understand your question.

Answer:

m<GEF = 66°

Step-by-step explanation:

(72+60)/2

= 132/2

= 66

Answered by GAUTHMATH

Answer:

csc = 6/5= 1.2

cot = √(11)/5= 0.6633

sin = 5÷6= 0.83333

Answer:

606.6 and 233.3 respectively

Step-by-step explanation:

Let the speed of plane in still air be x and the speed of wind be y.

ATQ, (x+y)*9=7560 and (x-y)*9=3360. Solving it, we get x=606.6 and y=233.3