Solving Polynomial Equations with Complex Solutions — Quiz

Question 5 • Algebra 2 with Probability

What are all the solutions to the equation 10 x to the 4th power plus 30 x cubed plus 35 x squared is equal to 0 ?

Answer

0 comma the fraction with numerator negative 3 plus i square root of 5 and denominator 2 comma the fraction with numerator negative 3 minus i square root of 5 and denominator 2

0 comma the fraction with numerator 3 plus i square root of 5 and denominator 2 comma the fraction with numerator 3 minus i square root of 5 and denominator 2

0 comma negative 6 plus 2 i square root of 5 comma negative 6 minus 2 i square root of 5

0 comma 6 plus 2 i square root of 5 comma 6 minus 2 i square root of 5

Explanation

Factor the equation: 10x⁴ + 30x³ + 35x² = 5x²(2x² + 6x + 7) = 0. This gives x = 0, and the quadratic equation 2x² + 6x + 7 = 0. Using the quadratic formula, its roots are (−6 ± √(36 − 56))/4 = (−3 ± i√5)/2, so the solutions match choice B.

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