1.Show roots of x^2+2x-(m^2-1)=0 are rational values of m. 2.The equation in terms of x, 2q(p+r)x+(p^2+q^2)x^2+(q^2+r^2)=0 has real roots.Show q^2=prPlease show each step carefully. Thanks.

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nathanshields | High School Teacher | (Level 1) Associate Educator

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1. Let's hit this bad boy with the quadratic formula


The only thing that would make x irrational is that square root.  So let's focus on that.

`sqrt(4+4(m^2-1)) = sqrt(4(1+m^2-1))=sqrt(4m^2)=2m`

How nice!


So the roots are not only rational multiples of m, but they are specifically m-1 and -m-1.



Let's use the quadratic formula, eh?

`x = (-2q(p+r)+-sqrt(4q^2(p+r)^2-4(p^2+q^2)(q^2+r^2)))/(2(p^2+q^2))`

We're told the function has real roots.  So the value under the square root must be positive (or zero).  So...


Cancel the 4s and multiply this stuff out



Subtract like quantities from both sides




A square number cannot be less than zero, so it must be equal to zero.


And I think you can take it from there.






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