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3 years ago

For any integer x, x squared -x will always produce an even value

Mathematics

2 answers:

wariber [46]3 years ago

6 0

Answer:

$(x^2 - x)$ will always have an even value.

Step-by-step explanation:

We are given an integer x.

Let x be even, then it can be written in the form x = 2n, where n is an integer.

If we evaluate,

$(x^2 - x) = (2n)^2 - 2n = 4n^2 - 2n = 2(2n^2 - n)$

Thus, it have an even value.

If we take x to be an odd integer, then,it can be written in the form x = 2n+1, where n is an integer.

$(x^2 - x) = (2n+1)^2 - 2n = 4n^2 + 2n = 2(2n^2 + n)$

Thus, it have an even value.

Thus,

$(x^2 - x)$ will always have an even value.

Alenkinab [10]3 years ago

3 0

Not at all:

The square of an even number will generate an even number and
The square of an odd number will generate an odd number

You might be interested in

The length of the base of an isosceles triangle is x. The length of a leg is 2x-6. The perimeter of the triangle is 23 . Find x.

expeople1 [14]

Answer:

In an isosceles triangle, the legs of the triangle have same lengths.

${ \tt{perimeter = side + side + side}}$

• substitute:

${ \tt{23 = x + 2(2x - 6)}} \\ \\ { \tt{23 = x + 4x - 12}} \\ \\ { \tt{5x = 35}} \\ \\ { \boxed{ \tt{ \: \: x = 7 \: \: }}}$

7 0

3 years ago

Sin(x)^4+ cos(x)^4=1/2

posledela

I'll assume that what was meant was $\sin ^4 x + \cos ^4 x = \dfrac{1}{2}$ .

The exponent in the funny place is just an abbreviation: $\sin ^4 x = (\sin x)^4$ .

I hope that's what you meant. Let me know if I'm wrong.

Let's start from the old saw

$\cos^2 x + \sin ^2x = 1$

Squaring both sides,

$(\cos^2 x + \sin ^2x)^2 = 1^2$

$\cos^4 x + 2 \cos ^2 x \sin ^2x +\sin ^4x = 1$

$\cos^4 x + \sin ^4x = 1 - 2 \cos ^2 x \sin ^2x$

So now the original question

$\sin ^4 x + \cos ^4 x = \dfrac{1}{2}$

becomes
$1 - 2 \cos ^2 x \sin ^2x = \dfrac{1}{2}$

$4 \cos ^2 x \sin ^2x = 1$

Now we use the sine double angle formula

$\sin 2x = 2 \sin x \cos x$

We square it to see

$\sin^2 2x = 4\sin^2 x \cos^2 x = 1$

Taking the square root,

$\sin 2x = \pm 1$

Not sure how you want it; we'll do it in degrees.

When we know the sine of an angle, there's usually two angles on the unit circle that have that sine. They're supplementary angles which add to $180^\circ$ . But when the sine is 1 or -1 like it is here, we're looking at $90^\circ$ and $-90^\circ$ , which are essentially their own supplements, slightly less messy.

That means we have two equations:

$\sin 2x = 1 = \sin 90^\circ$

$2x = 90^\circ + 360^\circ k \quad$ integer $k$

$x = 45^\circ + 180^\circ k$

or

$\sin 2x = -1 = \sin -90^\circ$

$2x = -90^\circ+ 360^\circ k$

$x = - 45^\circ + 180^\circ k$

We can combine those for a final answer,

$x = \pm 45^\circ + 180^\circ k \quad$ integer $k$

Check. Let's just check one, how about

$x=-45^\circ + 180^\circ = 135^\circ$

$\sin(135)= 1/\sqrt{2}$

$\sin ^4(135)=(1/\sqrt{2})^4 = 1/4$

$\cos ^4(135)=(-1/\sqrt{2})^4 = 1/4$

$\sin ^4(135^\circ) +\cos ^4(135^\circ) = 1/2 \quad\checkmark$

6 0

3 years ago

Make a frequency distribution and find the relative frequencies for the following number set. Round the relative frequency to th

Pani-rosa [81]

Solution: We have to find the Frequency and Relative frequency of the given data:

Frequency is the number of times a number occurs.

Relative Frequency is the number of times a number occurs divided by the total number of items.

Therefore, the frequency and relative frequency are calculated as below:

Number Frequency Relative Frequency

20 1 $\frac{1}{31} \times 100 =3.2\%$