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−755-75\sqrt{5} is a root of f(x)=x2−28,125f(x) = x^2 - 28,125. Find the other roots of f(x)f(x).\newlineWrite your answer as a list of simplified values separated by commas, if there is more than one value.

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Full solution

Q. −755-75\sqrt{5} is a root of f(x)=x2−28,125f(x) = x^2 - 28,125. Find the other roots of f(x)f(x).\newlineWrite your answer as a list of simplified values separated by commas, if there is more than one value.
  1. Identify Product of Roots: Since −755-75\sqrt{5} is a root, we can use the fact that the product of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 is ca\frac{c}{a}. Here, a=1a = 1 and c=−28,125c = -28,125.
  2. Let Other Root be rr: Let the other root be rr. Then, (−755)×r=−28,125(-75\sqrt{5}) \times r = -28,125.
  3. Solve for r: Solve for r: r=−28,125−755r = \frac{-28,125}{-75\sqrt{5}}.
  4. Simplify Expression: Simplify the expression: r=3755r = \frac{375}{\sqrt{5}}.
  5. Rationalize Denominator: To rationalize the denominator, multiply the numerator and denominator by 5\sqrt{5}: r=37555r = \frac{375\sqrt{5}}{5}.
  6. Final Simplification: Simplify the fraction: r=755r = 75\sqrt{5}.

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