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3 years ago

4(x+15) = 2(2x+25) has how many solutions?

Mathematics

2 answers:

ankoles [38]3 years ago

8 0

Answer:

Step-by-step explanation:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

4*(x-10)^2-(25)=0

Step by step solution :

STEP

Equation at the end of step 1

4 • (x - 10)2 - 25 = 0

STEP

2.1 Evaluate : (x-10)2 = x2-20x+100

Trying to factor by splitting the middle term

2.2 Factoring 4x2-80x+375

The first term is, 4x2 its coefficient is 4 .

The middle term is, -80x its coefficient is -80 .

The last term, "the constant", is +375

Step-1 : Multiply the coefficient of the first term by the constant 4 • 375 = 1500

Step-2 : Find two factors of 1500 whose sum equals the coefficient of the middle term, which is -80 .

-1500 + -1 = -1501

-750 + -2 = -752

-500 + -3 = -503

-375 + -4 = -379

-300 + -5 = -305

-250 + -6 = -256

-150 + -10 = -160

-125 + -12 = -137

-100 + -15 = -115

-75 + -20 = -95

-60 + -25 = -85

-50 + -30 = -80 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -50 and -30

4x2 - 50x - 30x - 375

Step-4 : Add up the first 2 terms, pulling out like factors :

2x • (2x-25)

Add up the last 2 terms, pulling out common factors :

15 • (2x-25)

Step-5 : Add up the four terms of step 4 :

(2x-15) • (2x-25)

Which is the desired factorization

Equation at the end of step

(2x - 25) • (2x - 15) = 0

STEP

Theory - Roots of a product

3.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

3.2 Solve : 2x-25 = 0

Add 25 to both sides of the equation :

2x = 25

Divide both sides of the equation by 2:

x = 25/2 = 12.500

Solving a Single Variable Equation:

3.3 Solve : 2x-15 = 0

Add 15 to both sides of the equation :

2x = 15

Divide both sides of the equation by 2:

x = 15/2 = 7.500

Supplement : Solving Quadratic Equation Directly

Solving 4x2-80x+375 = 0 directly

Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex:

4.1 Find the Vertex of y = 4x2-80x+375

Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 4 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 10.0000

Plugging into the parabola formula 10.0000 for x we can calculate the y -coordinate :

y = 4.0 * 10.00 * 10.00 - 80.0 * 10.00 + 375.0

or y = -25.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = 4x2-80x+375

Axis of Symmetry (dashed) {x}={10.00}

Vertex at {x,y} = {10.00,-25.00}

x -Intercepts (Roots) :

Root 1 at {x,y} = { 7.50, 0.00}

Root 2 at {x,y} = {12.50, 0.00}

Solve Quadratic Equation by Completing The Square

4.2 Solving 4x2-80x+375 = 0 by Completing The Square .

Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :

x2-20x+(375/4) = 0

Subtract 375/4 from both side of the equation :

x2-20x = -375/4

Now the clever bit: Take the coefficient of x , which is 20 , divide by two, giving 10 , and finally square it giving 100

Add 100 to both sides of the equation :

On the right hand side we have :

-375/4 + 100 or, (-375/4)+(100/1)

The common denominator of the two fractions is 4 Adding (-375/4)+(400/4) gives 25/4

So adding to both sides we finally get :

x2-20x+100 = 25/4

Adding 100 has completed the left hand side into a perfect square :

x2-20x+100 =

(x-10) • (x-10) =

(x-10)2

Things which are equal to the same thing are also equal to one another. Since

x2-20x+100 = 25/4 and

x2-20x+100 = (x-10)2

then, according to the law of transitivity,

(x-10)2 = 25/4

We'll refer to this Equation as Eq. #4.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(x-10)2 is

(x-10)2/2 =

(x-10)1 =

x-10

Two solutions were found :

x = 15/2 = 7.500

x = 25/2 = 12.500

IceJOKER [234]3 years ago

7 0

Answer: There are no solutions.

Step-by-step explanation: Hope this help :D

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3 years ago

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Find four numbers that form a geometric progression such that the third term is greater than the first by 12 and the fourth is g

olchik [2.2K]

If $x$ is the first number in the progression, and $r$ is the common ratio between consecutive terms, then the first four terms in the progression are

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We want to have

$\begin{cases}xr^2-x=12\\xr^3-xr=36\end{cases}$

In the second equation, we have

$xr^3-xr=xr(r^2-1)=36$

and in the first, we have

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3 years ago

In an examination of purchasing patterns of shoppers, a sample of 16 shoppers revealed that they spent, on average, $54 per hour

Lesechka [4]

Answer:

$54-1.64\frac{21}{\sqrt{16}}=45.39$

$54 +1.64\frac{21}{\sqrt{16}}=62.61$

So on this case the 90% confidence interval would be given by (45.39;62.61)

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X=54$ represent the sample mean

$\mu$ population mean (variable of interest)

$\sigma=21$ represent the sample standard deviation

n=16 represent the sample size

Solution to the problem

The confidence interval for the mean is given by the following formula:

$\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$ (1)

Since the Confidence is 0.90 or 90%, the value of $\alpha=1-0.9=0.1$ and $\alpha/2 =0.05$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that $z_{\alpha/2}=1.64$