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3 years ago

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what are the exact solutions of x^2 = 4 − 7x? x = x equals negative 7 plus or minus the square root of thirty-three all over 2 x

= x equals negative 7 plus or minus the square root of sixty-five all over 2 x = x equals 7 plus or minus the square root of sixty-five all over 2 x = x equals 7 plus or minus the square root of thirty-three all over 2

1 answer:

Archy [21]3 years ago

4 0

x² = -7x + 4
x² + 7x - 4 = 0
x = -7 ± √(7² - 4(1)(-4))
2(1)
x = -7 ± √(49 + 16)
2
x = -7 ± √(65)
2
x ≈ -7 ± 8.06
2
x ≈ -3.5 ± 4.03
x ≈ -3.5 + 4.03 or x ≈ -3.5 - 4.03
x ≈ 0.53 or x ≈ -7.53

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I need help with my math homework. The questions is: Find all solutions of the equation in the interval [0,2π).

Aleksandr-060686 [28]

Answer:

$\frac{7\pi}{24}$ and $\frac{31\pi}{24}$

Step-by-step explanation:

$\sqrt{3} \tan(x-\frac{\pi}{8})-1=0$

Let's first isolate the trig function.

Add 1 one on both sides:

$\sqrt{3} \tan(x-\frac{\pi}{8})=1$

Divide both sides by $\sqrt{3}$ :

$\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}$

Now recall $\tan(u)=\frac{\sin(u)}{\cos(u)}$ .

$\frac{1}{\sqrt{3}}=\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}$

or

$\frac{1}{\sqrt{3}}=\frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}}$

The first ratio I have can be found using $\frac{\pi}{6}$ in the first rotation of the unit circle.

The second ratio I have can be found using $\frac{7\pi}{6}$ you can see this is on the same line as the $\frac{\pi}{6}$ so you could write $\frac{7\pi}{6}$ as $\frac{\pi}{6}+\pi$ .

So this means the following:

$\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}$

is true when $x-\frac{\pi}{8}=\frac{\pi}{6}+n \pi$

where $n$ is integer.

Integers are the set containing {..,-3,-2,-1,0,1,2,3,...}.

So now we have a linear equation to solve:

$x-\frac{\pi}{8}=\frac{\pi}{6}+n \pi$

Add $\frac{\pi}{8}$ on both sides:

$x=\frac{\pi}{6}+\frac{\pi}{8}+n \pi$

Find common denominator between the first two terms on the right.

That is 24.

$x=\frac{4\pi}{24}+\frac{3\pi}{24}+n \pi$

$x=\frac{7\pi}{24}+n \pi$ (So this is for all the solutions.)

Now I just notice that it said find all the solutions in the interval $[0,2\pi)$ .

So if $\sqrt{3} \tan(x-\frac{\pi}{8})-1=0$ and we let $u=x-\frac{\pi}{8}$ , then solving for $x$ gives us:

$u+\frac{\pi}{8}=x$ ( I just added $\frac{\pi}{8}$ on both sides.)

So recall $0\le x$ .

Then $0 \le u+\frac{\pi}{8}$ .

Subtract $\frac{\pi}{8}$ on both sides:

$-\frac{\pi}{8}\le u$

Simplify:

$-\frac{\pi}{8}\le u$

$-\frac{\pi}{8}\le u$

So we want to find solutions to:

$\tan(u)=\frac{1}{\sqrt{3}}$ with the condition:

$-\frac{\pi}{8}\le u$

That's just at $\frac{\pi}{6}$ and $\frac{7\pi}{6}$

So now adding $\frac{\pi}{8}$ to both gives us the solutions to:

$\tan(x-\frac{\pi}{8})=\frac{1}{\sqrt{3}}$ in the interval:

$0\le x$ .

The solutions we are looking for are:

$\frac{\pi}{6}+\frac{\pi}{8}$ and $\frac{7\pi}{6}+\frac{\pi}{8}$

Let's simplifying:

$(\frac{1}{6}+\frac{1}{8})\pi$ and $(\frac{7}{6}+\frac{1}{8})\pi$

$\frac{7}{24}\pi$ and $\frac{31}{24}\pi$

$\frac{7\pi}{24}$ and $\frac{31\pi}{24}$

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While preparing for a party, Luis spent 25 minutes blowing up 10 balloons. Write an equation to express how many minutes it took

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What is true about the slope of the function below<br> y = 4x + 3

Karo-lina-s [1.5K]

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$\blue{\textsf{\textbf{\underline{\underline{Question:-}}}}}$

Which is true about the slope of the function y=4x+3?

$\blue\textsf{\textbf{\underline{\underline{Answer and How to Solve:-}}}}$

The slope of the function y=4x+3 is 4; in equations like this one, the slope is simply the number before x.

The rise is 4, and the run is 1.

Remember, slope is also defined as "Rise/Run"

[Note:- You haven't listed any options here, could you kindly add them in the comments?]

<h3>Good luck.</h3>

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2 years ago

Which are correct representations of the inequality –3(2x – 5) < 5(2 – x)? Select two options.

chubhunter [2.5K]

The correct statements are options C and option D.

- 6x + 15 < 10 - 5x ⇒ 3rd answer

An open circle is at 5 and a bold line starts at 5 and is pointing to the right.

<h3 /><h3>What is inequality?</h3>

The relation between two expressions that are not equal, employing a sign such as ≠ ‘not equal to’, > ‘greater than, or < ‘less than.

The inequality is -3(2x - 5) < 5(2 - x)

At first, simplify each side

-3(2x - 5) = -3(2x) + -3(-5)

Remember (-)(-) = (+)

-3(2x - 5) = - 6x + 15

5(2 - x) = 5(2) + 5(-x)

Remember (+)(-) = (-)

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Subtract 15 from both sides

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Add 5x to both sides

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Remember the coefficient of x is negative, then when you divide both sides by it you must reverse the sign of inequality

The coefficient of x is -1

Divide both sides by -1

x > 5

Therefore the correct statements are options C and option D.

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An open circle is at 5 and a bold line starts at 5 and is pointing to the right.

To know more about inequality follow

brainly.com/question/24372553

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2 years ago