Finding Roots of Quadratic Equations & Sinusoidal Functions

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In summary: Thanks for trying to explain it!I don't understand how you got those angles. Can you please elaborate?
  • #1
eleventhxhour
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4) Consider the equation H(t) = 16(2)^2t - 10(2)^t + 1. What are its roots? (HINT: Does this look like a quadratic? Perhaps, at least at first, it should be treated like one).

5) Do the same for Y(x) = 2sin^2x - 3sinx - 2. What is wrong with your solutions?

Even with the hint, I'm not really sure how to start this, especially with the variables in the exponents...Could someone help? Thanks
 
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  • #2
For the first equation, first write it as:

\(\displaystyle 16\left(2^t\right)^2-10(2)^t+1=0\)

Next, you may use a substitution such as:

\(\displaystyle u=2^t\)

And you may now write:

\(\displaystyle 16u^2-10u+1=0\)

Now solve for $u$, and once you have the roots, then back-substitute for $u$ and solve for $t$.

The other problem may be worked the same way.
 
  • #3
This is what I did:
Factored it: (8u-1)(2u-1) = 0
and then found the zeros: 1, 1/8 and 1/2

And then I'm not sure about the next part. Do I substitute 2^t for u?
[8(2^t)-1][2(2^t)-1] = 0
(16^t - 1)(4^t-1)

and then how would you find the roots of that? (assuming that I did that correctly...)

Thanks again!
 
  • #4
You factored correctly, but from where did the root $u=1$ come? Let's look at the root:

\(\displaystyle u=\frac{1}{2}\)

Now back substitute for $u$:

\(\displaystyle 2^t=\frac{1}{2}=2^{-1}\)

Equate exponents:

$t=-1$

Can you now do the same for the root \(\displaystyle u=\frac{1}{8}\) ?
 
  • #5
MarkFL said:
You factored correctly, but from where did the root $u=1$ come? Let's look at the root:

\(\displaystyle u=\frac{1}{2}\)

Now back substitute for $u$:

\(\displaystyle 2^t=\frac{1}{2}=2^{-1}\)

Equate exponents:

$t=-1$

Can you now do the same for the root \(\displaystyle u=\frac{1}{8}\) ?

Hm...would it be 2^-3 for the root u = 1/8?
 
  • #6
Recall, we used the substitution \(\displaystyle u=2^t\), so now we are putting $2^t$ back in, in place of $u$.

Also recall:

\(\displaystyle \frac{1}{a^b}=a^{-b}\)

Hence:

\(\displaystyle \frac{1}{2}=\frac{1}{2^1}=2^{-1}\)
 
  • #7
MarkFL said:
Recall, we used the substitution \(\displaystyle u=2^t\), so now we are putting $2^t$ back in, in place of $u$.

Also recall:

\(\displaystyle \frac{1}{a^b}=a^{-b}\)

Hence:

\(\displaystyle \frac{1}{2}=\frac{1}{2^1}=2^{-1}\)

Sorry, I edited my previous reply. I think I understand now. Would 2^-3 be correct?
 
  • #8
Yes, good! :D

\(\displaystyle \frac{1}{8}=\frac{1}{2^3}=2^{-3}\)
 
  • #9
MarkFL said:
Yes, good! :D

\(\displaystyle \frac{1}{8}=\frac{1}{2^3}=2^{-3}\)

So does that mean the roots are t=-1 and t=-3?

And for the second part of the question, this is what I've done so far:

5) Do the same for Y(x) = 2sin^2x - 3sinx - 2. What is wrong with your solutions?

let sinx = u, so...
y(x) = 2u^2 - 3u - 2
(u - 2)(2u + 1)
u = 2, u =-1/2

but I'm not sure what to do after that...

Thanks again! (:
 
Last edited:
  • #10
What is the range of the sine function?
 
  • #11
MarkFL said:
What is the range of the sine function?

Can't they only range from -1 to 1?
 
  • #12
Correct, so you must discard the root:

\(\displaystyle \sin(x)=2\)

Because no real value for $x$ will satisfy that equation. So, you are left with:

\(\displaystyle \sin(x)=-\frac{1}{2}\)

In which quadrants will the solutions reside?
 
  • #13
MarkFL said:
Correct, so you must discard the root:

\(\displaystyle \sin(x)=2\)

Because no real value for $x$ will satisfy that equation. So, you are left with:

\(\displaystyle \sin(x)=-\frac{1}{2}\)

In which quadrants will the solutions reside?

\(\displaystyle \sin(x)=\frac{y}{r}\)

$y = -1$, $r = 2$

So since $y$ is negative, it'd reside in either quadrant 3 or 4.

Is this correct?
 
  • #14
Yes, the sine function is negative in the 3rd and 4th quadrants. I would try to write all solutions using one statement. In degrees, we should observe that we may write the two angles as:

\(\displaystyle \theta=270^{\circ}\pm60^{\circ}\)

Notice I picked the angle midway between the two solutions, and then used $\pm$ to pick both angles up with one statement. Now, what must we add to this to get all the solutions...Hint: think of the periodicity of the sine function. :D
 
  • #15
MarkFL said:
Yes, the sine function is negative in the 3rd and 4th quadrants. I would try to write all solutions using one statement. In degrees, we should observe that we may write the two angles as:

\(\displaystyle \theta=270^{\circ}\pm60^{\circ}\)

Notice I picked the angle midway between the two solutions, and then used $\pm$ to pick both angles up with one statement. Now, what must we add to this to get all the solutions...Hint: think of the periodicity of the sine function. :D

Hm, I don't really understand how you got those angles?
 
  • #16
eleventhxhour said:
Hm, I don't really understand how you got those angles?

When I think of $\sin(x)=-\frac{1}{2}$, I picture the unit circle and the line $y=-\frac{1}{2}$. I then see that the two points of intersection between the two occur at $\theta=210^{\circ},\,330^{\circ}$, and I can see that $210^{\circ}=270^{\circ}-60^{\circ}$ and $330^{\circ}=270^{\circ}+60^{\circ}$. This allows us to write both angles in one statement:

\(\displaystyle \theta=270^{\circ}\pm60^{\circ}\)

There are of course other ways to think of it, but this is how I do it. :D
 
  • #17
MarkFL said:
When I think of $\sin(x)=-\frac{1}{2}$, I picture the unit circle and the line $y=-\frac{1}{2}$. I then see that the two points of intersection between the two occur at $\theta=210^{\circ},\,330^{\circ}$, and I can see that $210^{\circ}=270^{\circ}-60^{\circ}$ and $330^{\circ}=270^{\circ}+60^{\circ}$. This allows us to write both angles in one statement:

\(\displaystyle \theta=270^{\circ}\pm60^{\circ}\)

There are of course other ways to think of it, but this is how I do it. :D

Ohh okay, that makes sense. Though I'm not sure what to add to get all the solutions?
 
  • #18
You were not given any restrictions on $x$, so use the following:

\(\displaystyle \sin\left(x+k\cdot360^{\circ}\right)=\sin(x)\) where $k\in\mathbb{Z}$ (this means $k$ can be any integer).

In other words, moving around the unit circle any number of complete revolutions in either direction will put you right back in the same place on the circle, and the sine of that new angle will be the same as the original.
 

FAQ: Finding Roots of Quadratic Equations & Sinusoidal Functions

What is the general form of a quadratic equation?

The general form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are constants and x is the variable.

How do you find the roots of a quadratic equation?

The roots of a quadratic equation can be found by using the quadratic formula: x = (-b ± √(b2 - 4ac)) / 2a. Alternatively, you can also factor the equation to find the roots.

What is the discriminant of a quadratic equation and how is it used to determine the nature of the roots?

The discriminant of a quadratic equation is the expression b2 - 4ac. It is used to determine the nature of the roots as follows:

  • If the discriminant is greater than 0, the equation has two distinct real roots.
  • If the discriminant is equal to 0, the equation has a single real root.
  • If the discriminant is less than 0, the equation has two complex roots.

What are sinusoidal functions?

Sinusoidal functions are mathematical functions that model periodic phenomena, such as sound waves, light waves, and the motion of objects on a spring. They can be represented by the general form y = a sin(bx + c) + d, where a is the amplitude, b is the frequency, c is the phase shift, and d is the vertical shift.

How do you find the period and amplitude of a sinusoidal function?

The period of a sinusoidal function is the length of one full cycle of the function, and it can be found by using the formula T = 2π/b, where b is the frequency. The amplitude is the distance between the center line and the maximum or minimum value of the function, and it can be found by taking half the difference between the maximum and minimum values of the function.

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