To convert 45000 inches into miles which two ratios could you multiply by

Any one know how to find the roots of an equation?

sergiy2304 [10]

Solutions or Roots of Quadratic Equations. A real number x will be called a solution or a root if it satisfies the equation, meaning . It is easy to see that the roots are exactly the x-intercepts of the quadratic function , that is the intersection between the graph of the quadratic function with the x-axis.

4 0

3 years ago

Solve the following equations: (a) x^11=13 mod 35 (b) x^5=3 mod 64

tino4ka555 [31]

a.

$x^{11}=13\pmod{35}\implies\begin{cases}x^{11}\equiv13\equiv3\pmod5\\x^{11}\equiv13\equiv6\pmod7\end{cases}$

By Fermat's little theorem, we have

$x^{11}\equiv (x^5)^2x\equiv x^3\equiv3\pmod5$

$x^{11}\equiv x^7x^4\equiv x^5\equiv6\pmod 7$

5 and 7 are both prime, so $\varphi(5)=4$ and $\varphi(7)=6$ . By Euler's theorem, we get

$x^4\equiv1\pmod5\implies x\equiv3^{-1}\equiv2\pmod5$

$x^6\equiv1\pmod7\impleis x\equiv6^{-1}\equiv6\pmod7$

Now we can use the Chinese remainder theorem to solve for $x$ . Start with

$x=2\cdot7+5\cdot6$

Taken mod 5, the second term vanishes and $14\equiv4\pmod5$ . Multiply by the inverse of 4 mod 5 (4), then by 2.

$x=2\cdot7\cdot4\cdot2+5\cdot6$

Taken mod 7, the first term vanishes and $30\equiv2\pmod7$ . Multiply by the inverse of 2 mod 7 (4), then by 6.

$x=2\cdot7\cdot4\cdot2+5\cdot6\cdot4\cdot6$

$\implies x\equiv832\pmod{5\cdot7}\implies\boxed{x\equiv27\pmod{35}}$

b.

$x^5\equiv3\pmod{64}$

We have $\varphi(64)=32$ , so by Euler's theorem,

$x^{32}\equiv1\pmod{64}$

Now, raising both sides of the original congruence to the power of 6 gives

$x^{30}\equiv3^6\equiv729\equiv25\pmod{64}$

Then multiplying both sides by $x^2$ gives

$x^{32}\equiv25x^2\equiv1\pmod{64}$

so that $x^2$ is the inverse of 25 mod 64. To find this inverse, solve for $y$ in $25y\equiv1\pmod{64}$ . Using the Euclidean algorithm, we have

64 = 2*25 + 14

25 = 1*14 + 11

14 = 1*11 + 3

11 = 3*3 + 2

3 = 1*2 + 1

=> 1 = 9*64 - 23*25

so that $(-23)\cdot25\equiv1\pmod{64}\implies y=25^{-1}\equiv-23\equiv41\pmod{64}$ .

So we know

$25x^2\equiv1\pmod{64}\implies x^2\equiv41\pmod{64}$

Squaring both sides of this gives

$x^4\equiv1681\equiv17\pmod{64}$

and multiplying both sides by $x$ tells us

$x^5\equiv17x\equiv3\pmod{64}$

Use the Euclidean algorithm to solve for $x$ .

64 = 3*17 + 13

17 = 1*13 + 4

13 = 3*4 + 1

=> 1 = 4*64 - 15*17

so that $(-15)\cdot17\equiv1\pmod{64}\implies17^{-1}\equiv-15\equiv49\pmod{64}$ , and so $x\equiv147\pmod{64}\implies\boxed{x\equiv19\pmod{64}}$

5 0

3 years ago

An electrician charges $40 per visit, and $20 per hour of work. On a particular day, he made 3 visits and calculated that his av

saw5 [17]

Answer:

4 hours

Step-by-step explanation:

For three visits, the electrician earned $40 × 3 = $120 in "per-visit" charges. For working x hours, he earned 20x in "per-hour" charges. The total of these came to 50x:

120 +20x = 50x

120 = 30x . . . . . . . . subtract 20x

4 = x . . . . . . . . . . . . . divide by 30

The electrician worked 4 hours that day.

3 0

3 years ago

To add 20.4 + 3.75 + 4.25, Crystal first added 3.75 and 4.25. Which property did Crystal use to add mentally?

evablogger [386]

Answer:

Associative Property of Addition

Step-by-step explanation:

From the list of given options, option A correctly answers the question and this is because

$a + b + c = a + (b + c)$ --- (1)

or

$a + b + c = (a + b) + c$ --- (2)

The above illustrations only apply to Associative Property of Addition

In Crystal's case:

$20.4 + 3.75 + 4.25 = 20.4 + (3.75 + 4.25)$

This can be compared to (1) above

Hence;

Option A answers the question

3 0

3 years ago

Simpifly (-5x2 - 3x - 7) + (-2x3 + 6x2 - 8)

ioda

Answer:

-2x³ + x² - 3x - 15

Step-by-step explanation:

Simply combine like terms together:

-5x² - 3x - 7 - 2x³ + 6x² - 8

-2x³ + (-5x² + 6x²) - 3x + (-7 - 8)

-2x³ + x² - 3x + (-7 - 8)

-2x³ + x² - 3x - 15

5 0

3 years ago

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