find the nature of the roots of the roots of the equation (x-705)(x-795)+800(x-750)(x-835)=0.

(A) Imaginary roots.
(B) Real and distinct.
(C) Real and equal.
(D) None.

If you are allowed to use a graphing calculator, this can solve a lot of time. Graphing y = (x - 705)(x - 795) + 800(x - 750)(x - 835) gets us a parabola with x-intercepts of 749.97 and 834.924.

These roots are real and distinct, or choice B.

If you aren't allowed to use a graphing calculator, you can expand and combine like terms to get 801x^2 − 1269500x + 501560475. Looking at the discriminant, we see that (−1269500)^2 - 4*801*501560475 = 4630488100 is greater than 0 and not a perfect square, so it is real and distinct, or choice B.

I hope that helps!

find the nature of the roots of the roots of the equation (x-705)(x-795)+800(x-750)(x-835)=0.(A) Imaginary roots.(B) Real and distinct.(C) Real and equal.(D) None.​

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find the nature of the roots of the roots of the equation (x-705)(x-795)+800(x-750)(x-835)=0.

(A) Imaginary roots.
(B) Real and distinct.
(C) Real and equal.
(D) None.