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2 years ago

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EXAMPLE 10 Show that there is a root of the equation 2x3 − 4x2 + 3x − 2 = 0 between 1 and 2. SOLUTION Let f(x) = 2x3 − 4x2 + 3x

− 2 = 0. We are looking for a solution of the given equation, that is, a number c between 1 and 2 such that f(c) = . Therefore we take a = , b = , and N = in the Intermediate Value Theorem. We have

1 answer:

muminat2 years ago

3 0

Answer: $c = 1.2$

Step-by-step explanation:

First, it is needed to determined the values for x = 1 and x = 2:

$f(1) = -1, f(2)=4$

The sign change within the interval is the most sound evidence of the root existence. According to the Intermediate Value Theorem, there is a number $c$ such that $f(c) = 0$ . Another finding is that $c$ is closer to 1 than to 2.

$c = a + \frac{b-a}{f(b)-f(a)}\cdot[f(c)-f(a)]$

$c = 1 + \frac{2-1}{4-(-1)}\cdot[0-(-1)] \\c = 1.2$

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Show work please<br> \sqrt(x+12)-\sqrt(2x+1)=1

Nesterboy [21]

Answer:

$x=4$

Step-by-step explanation:

Given $\displaystyle\\\sqrt{x+12}-\sqrt{2x+1}=1$ , start by squaring both sides to work towards isolating $x$ :

$\displaystyle\\\left(\sqrt{x+12}-\sqrt{2x+1}\right)^2=\left(1\right)^2$

Recall $(a-b)^2=a^2-2ab+b^2$ and $\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$ :

$\displaystyle\\\left(\sqrt{x+12}-\sqrt{2x+1}\right)^2=\left(1\right)^2\\\implies x+12-2\sqrt{(x+12)(2x+1)}+2x+1=1$

Isolate the radical:

$\displaystyle\\x+12-2\sqrt{(x+12)(2x+1)}+2x+1=1\\\implies -2\sqrt{(x+12)(2x+1)}=-3x-12\\\implies \sqrt{(x+12)(2x+1)}=\frac{-3x-12}{-2}$

Square both sides:

$\displaystyle\\(x+12)(2x+1)=\left(\frac{-3x-12}{-2}\right)^2$

Expand using FOIL and $(a+b)^2=a^2+2ab+b^2$ :

$\displaystyle\\2x^2+25x+12=\frac{9}{4}x^2+18x+36$

Move everything to one side to get a quadratic:

$\displaystyle-\frac{1}{4}x^2+7x-24=0$

Solving using the quadratic formula:

A quadratic in $ax^2+bx+c$ has real solutions $\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ . In $\displaystyle-\frac{1}{4}x^2+7x-24$ , assign values:

$\displaystyle \\a=-\frac{1}{4}\\b=7\\c=-24$

Solving yields:

$\displaystyle\\x=\frac{-7\pm \sqrt{7^2-4\left(-\frac{1}{4}\right)\left(-24\right)}}{2\left(-\frac{1}{4}\right)}\\\\x=\frac{-7\pm \sqrt{25}}{-\frac{1}{2}}\\\\\begin{cases}x=\frac{-7+5}{-0.5}=\frac{-2}{-0.5}=\boxed{4}\\x=\frac{-7-5}{-0.5}=\frac{-12}{-0.5}=24 \:(\text{Extraneous})\end{cases}$

Only $x=4$ works when plugged in the original equation. Therefore, $x=24$ is extraneous and the only solution is $\boxed{x=4}$

4 0

2 years ago

Pls help<br><br> Given: △ABC, CM⊥ AB, BC = 5, AB = 7<br> CA = 4 sqrt(2)<br> Find: CM

babymother [125]

Answer:

The general plan is to find BM and from that CM. You need 2 equations to do that.

Step One

Set up the two equations.

(7 - BM)^2 + CM^2 = (4*sqrt(2) ) ^ 2 = 32

BM^2 + CM^2 = 5^2 = 25

Step Two

Subtract the two equations.

(7 - BM)^2 + CM^2 = 32

BM^2 + CM^2 = 25

(7 - BM)^2 - BM^2 = 7 (3)

Step three

Expand the left side of the new equation labeled (3)

49 - 14BM + BM^2 - BM^2 = 7

Step 4

Simplify And Solve

49 - 14BM = 7 Subtract 49 from both sides.

-49 - 14BM = 7 - 49

- 14BM = - 42 Divide by - 14

BM = -42 / - 14

BM = 3

Step Five

Find CM

CM^2 + BM^2 = 5^2

CM^2 + 3^2 = 5^2 Subtract 3^2 from both sides.

CM^2 = 25 - 9

CM^2 = 16 Take the square root of both sides.

sqrt(CM^2) = sqrt(16)

CM = 4 < Answer

Step-by-step explanation:

6 0

2 years ago

Factor the linear expression 16p- 24 using the greatest common factor of

horsena [70]

The Answer should be A

8 0

3 years ago

Read 2 more answers

A train travels 16 miles in 20 minutes At this rate , how many minutes will it take to trace 12 miles?

VARVARA [1.3K]

Answer:

Write and solve an equation of ratios:

16 miles 20 min

----------- = -----------

12 miles x

Then 16x = 240, and x = 240/16 = 15.

It will take 15 minutes to travel 12 miles.

3 0

2 years ago

The volume of a cube with sides of length s is given by v = s^3. find the rate of change of the volume with respect to s when s

katrin2010 [14]

The rate of change of the volume with respect to s when s = 15 centimeters is 3375 centimeters

<h3>What is the volume?</h3>

The formula for volume of a cube is given as;

Volume = a³

Where:

a = length of its side

From the information given, we have;

V = s³

Where s = 15 centimeters

Substitute the value into the formula

v = (15)³

v = 3375 centimeters

Thus, the rate of change of the volume with respect to s when s = 15 centimeters is 3375 centimeters

Learn more about a cube here:

brainly.com/question/1972490

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5 0

1 year ago