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Darkbouncer4689

Reducing quadratic equation (a=1)

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Hey all,

So for the quadratic equation x = (-b +- sqrt(b^2 - 4ac))/2a
when a=1, real-time rendering says it reduces to -b +- sqrt(b^2 - c)

Embarassingly enough I'm failing at basic algebra to figure out how they got that.

Could anyone show me the equalities to get to the fully reduced equation?

Thanks!

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Probably the ray/sphere hit test equation 13.11 in the 2nd edition of the book is meant:
t[sup]2[/sup] + 2 * b * t + c = 0

The quadratic equation
t[sup]2[/sup] + p * t + q = 0
has the solutions
-p/2 + sqrt( p[sup]2[/sup] / 4 - q )
-p/2 - sqrt( p[sup]2[/sup] / 4 - q )

If one compares that with the equation 13.11 in the book, then the relations
p := 2 * b
q := c
can be extracted, and hence the solutions are
-b + sqrt( b[sup]2[/sup] - c )
-b - sqrt( b[sup]2[/sup] - c )
what matches equation 13.12 in the book. Hence it seems me okay. You probably haven't chosen the correct substitutions.

EDIT: I just saw that wikipedia uses 3 coefficients for a quadratic equation:
a * t[sup]2[/sup] + b * t + c = 0
Well, first notice that the coefficient a, b, and c here are different from the b and c used in the book. Then divide the equation by the local a (assuming that it isn't zero)
t[sup]2[/sup] + b/a * t + c/a = 0
and substitute
p := b/a
q := c/a
and you yield in the simplified version I've used above:
t[sup]2[/sup] + p * t + q = 0

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Indeed it was the ray/sphere intersection. I had gotten to -b/2 +- sqrt(b^2/4 - c) and was scratching my head at how the heck they got to their equation, but I see now that I was forgetting that its 2tb, completely missed that.
Thanks for clearing that up! =)


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