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3 years ago

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(i) Write the expansion of (x + y)² and (x - y)². (ii) Find (x + y)² - (x - y)² (iii) Write 12 as the difference of two perfect

square.

1 answer:

zaharov [31]3 years ago

7 0

Answer:

1a (x + y)² = x² + 2xy + y²

1b. (x - y)² = x² - 2xy + y²

2. (x + y)² - (x - y)² = 4xy

3. 4² – 2² = 12

Step-by-step explanation:

1a. Expansion of (x + y)²

(x + y)² = (x + y)(x + y)

(x + y)² = x(x + y) + y(x + y)

(x + y)² = x² + xy + xy + y²

(x + y)² = x² + 2xy + y²

1b. Expansion of (x - y)²

(x - y)² = (x - y)(x - y)

(x - y)² = x(x - y) - y(x - y)

(x - y)² = x² - xy - xy + y²

(x - y)² = x² - 2xy + y²

2. Determination of (x + y)² - (x - y)²

This can be obtained as follow

(x + y)² = x² + 2xy + y²

(x - y)² = x² - 2xy + y²

(x + y)² - (x - y)² = x² + 2xy + y² - (x² - 2xy + y²)

= x² + 2xy + y² - x² + 2xy - y²

= x² - x² + 2xy + 2xy + y² - y²

= 2xy + 2xy

= 4xy

(x + y)² - (x - y)² = 4xy

3. Writing 12 as the difference of two perfect square.

To do this, we shall subtract 12 from a perfect square to obtain a number which has a perfect square root.

We'll begin by 4

4² – 12

16 – 12 = 4

Find the square root of 4

√4 = 2

4 has a square root of 2.

Thus,

4² – 12 = 4

4² – 12 = 2²

Rearrange

4² – 2² = 12

Therefore, 12 as a difference of two perfect square is 4² – 2²

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Simplify the following rational expression and express in expanded form.

VladimirAG [237]

Answer:

D m = $-\frac{3}{8}$

Step-by-step explanation:

factor the expressions in the denominator and the numerator to simplify the expression:

=> $\frac{2(-2m + 1)(2m - 1)}{2(2m - 1)(8m + 3)}$

=> $-\frac{2m + 1}{8m + 3}$

to make a fraction undefined, the numerator should be 0, thus, we substitute the values of m from the options into the denominator to make the denominator equals to 0:

=> $-\frac{2m + 1}{8(-\frac{3}{8} ) + 3}$ = $-\frac{2m + 1}{-3 + 3}$ = $-\frac{2m + 1}{0}$

in this case, the values of m from option D make the denominator of the fraction equals 0.

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2 years ago

Function f is an exponential function. It predicts the value of a famous painting, in thousands of dollars, as a function of the

nalin [4]

Answer:

$y=8 \cdot (\frac{5}{4})^x$

$f(x)=8 \cdot (\frac{5}{4})^x$

or

$f(x)=8 \cdot (1.25)^x$

Step-by-step explanation:

We are going to see if the exponential curve is of the form:

$y=a \cdot b^x$ , ( $b\neq 0$ ).

If you are given the $y-$ intercept, then $a$ is easy to find.

It is just the $y-$ coordinate of the $y-$ intercept is your value for $a$ .

(Why? The $y-$ intercept happens when $x=0$ . Replacing $x$ with 0 gives $y=a \cdot b^0=a \cdot 1=a$ . This says when $x=0 \text{ that} y=a$ .)

So $a=8$ .

So our function so far looks like this:

$y=8 \cdot b^x$

Now to find $b$ we need another point. We have two more points. So we will find $b$ using one of them and verify for our resulting equation works for the other.

Let's do this.

We are given $(1,10)$ is a point on our curve.

So when $x=1$ , $y=10$ .

$10=8 \cdot b^1$

$10=8 \cdot b$

Divide both sides by 8:

$\frac{10}{8}=b$

Reduce the fraction:

$\frac{5}{4}=b$

So the equation if it works out for the other point given is:

$y=8 \cdot (\frac{5}{4})^x$

Let's try it. So the last point given that we need to satisfy is $(2,12.5)$ .

This says when $x=2$ , $y=12.5$ .

Let's replace $x$ with 2 and see what we get for $y$ :

$y=8 \cdot (\frac{5}{4})^2$

$y=8 \cdot \frac{25}{16}$

$y=\frac{8}{16} \cdot 25$

$y=\frac{1}{2} \cdot 25}$

$y=\frac{25}{2}$

$y=12.5$

So we are good. We have found an equation satisfying all 3 points given.

The equation is $y=8 \cdot (\frac{5}{4})^x$ .

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3 years ago

Which of these statements is true for f(x)=(1/10)^x

lana66690 [7]

Step-by-step explanation:

Considering the function

$f\left(x\right)=\:\left(\frac{1}{10}\right)^x$

Analyzing option A)

Considering the function

$f\left(x\right)=\:\left(\frac{1}{10}\right)^x$

Putting $x = 1$ in the function

$f\left(1\right)=\:\left(\frac{1}{10}\right)^1$

$f\left(1\right)=\:\left\frac{1}{10}\right$

So, it is TRUE that when $x = 1$ then the out put will be $f\left(1\right)=\:\left\frac{1}{10}\right$

Therefore, the statement that '' The graph contains $\left(1,\:\frac{1}{10}\right)$ '' is TRUE.

Analyzing option B)

Considering the function

$f\left(x\right)=\:\left(\frac{1}{10}\right)^x$

The range of the function is the set of values of the dependent variable for which a function is defined.

$\mathrm{The\:range\:of\:an\:exponential\:function\:of\:the\:form}\:c\cdot \:n^{ax+b}+k\:\mathrm{is}\:\:f\left(x\right)>k$

$k=0$

$f\left(x\right)>0$

Thus,

$\mathrm{Range\:of\:}\left(\frac{1}{10}\right)^x:\quad \begin{bmatrix}\mathrm{Solution:}\:&\:f\left(x\right)>0\:\\ \:\mathrm{Interval\:Notation:}&\:\left(0,\:\infty \:\right)\end{bmatrix}$

Therefore, the statement that ''The range of $f(x)$ is $y > \frac{1}{10}$ " is FALSE

Analyzing option C)

Considering the function

$f\left(x\right)=\:\left(\frac{1}{10}\right)^x$

The domain of the function is the set of input values which the function is real and defined.

As the function has no undefined points nor domain constraints.

So, the domain is $-\infty \:$

Thus,

$\mathrm{Domain\:of\:}\:\left(\frac{1}{10}\right)^x\::\quad \begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:$

Therefore, the statement that ''The domain of $f(x)$ is $x>0$ '' is FALSE.

Analyzing option D)

Considering the function

$f\left(x\right)=\:\left(\frac{1}{10}\right)^x$

As the base of the exponential function is less then 1.

i.e. 0 < b < 1

Thus, the function is decreasing

Also check the graph of the function below, which shows that the function is decreasing.

Therefore, the statement '' It is always increasing '' is FALSE.

Keywords: function, exponential function, increasing function, decreasing function, domain, range

Learn more about exponential function from brainly.com/question/13657083

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$A=24^{2}$

A=576

The area of this square is 576cm2.

Hope this helps!

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2 years ago

Please help me quickly.<br> Thank you

Ludmilka [50]

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Hope it helps : )

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1 year ago

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