Online Course Discussion Forum

MCIII Algebra 5.8

 
 
Picture of Tina Jin
MCIII Algebra 5.8
by Tina Jin - Tuesday, 23 January 2024, 11:57 AM
 

Hi,

I don't understand the solution at the back of the book. Technically, the coefficient of x isn't 2(sin(pix/2)), since there's still a variable, x, in the coefficent, which is a contradivtion.

How do I solve the problem then?

Thanks

TIna Jin

 
Picture of Dr. Kevin Wang
Re: MCIII Algebra 5.8
by Dr. Kevin Wang - Tuesday, 23 January 2024, 11:38 PM
 

The question is: Find all real roots for the equation $$x^2 - 2x\sin\dfrac{\pi x}{2} + 1 = 0.$$

The solution treats this as a quadratic equation in $x$, and uses the discriminant.  There is no contradiction here.  As long as $x$ is a root, the inequality involving the discriminant is true even though $x$ appears in the coefficients.  Granted, this is not really a quadratic equation, but suppose $x=a$ is a real root to the original equation, then the following quadratic equation

$$x^2 - 2\left(\sin\dfrac{\pi a}{2}\right)x + 1 = 0$$

should also have a solution $x=a$.  The discriminant analysis works for this quadratic equation.  Essentially, there is no difference from the original method.

Picture of Tina Jin
Re: MCIII Algebra 5.8
by Tina Jin - Thursday, 25 January 2024, 7:31 AM
 

I see, so does the discriminant ALWAYS work? Even when the coefficient of it has an x in it?

For example, would this work:

x^2+(F(x))x+n=0, discriminant is (F(x))^2-4n.

Thanks,

Tina

Picture of Dr. Kevin Wang
Re: MCIII Algebra 5.8
by Dr. Kevin Wang - Thursday, 25 January 2024, 12:42 PM
 

We need to be specific in saying the following:  if the equation $x^2+(F(x))x+n=0$ has a real root for $x$, then $(F(x))^2-4n\geq 0$ for the same value of $x$.

Picture of Tina Jin
Re: MCIII Algebra 5.8
by Tina Jin - Monday, 29 January 2024, 7:43 AM
 
Thank you!