Question

Solve the equation for all complex and real solutions using the techniques from this module showing each step.

5x⁴+7x³+119x²+175x-150=0

Answer 1

To solve the given equation, 5x⁴+7x³+119x²+175x-150=0, we can factorize it into two quadratic equations and solve each separately. The solutions are x = x₁, x₂ from the first quadratic equation, and x = -3, x = 1 from the second quadratic equation.

Step-by-step explanation:

1. Given equation: 5x⁴+7x³+119x²+175x-150=0
2. One approach for solving this equation is to use the Rational Root Theorem to find the possible rational roots. However, since complex solutions are also requested, we will use another method.
3. Since the degree of the equation is 4, we can factor it into two quadratic equations and solve each separately. Let's set the equation equal to zero and then factorize it: 5x⁴+7x³+119x²+175x-150 = (x²-5x+10)(5x²+12x-15) = 0
4. Now, we can solve each quadratic equation separately. For the first equation, we can use the quadratic formula to find the solutions x₁ and x₂. For the second equation, we can factorize it further as (x+3)(5x-5) = 0 and find the solutions.
5. Therefore, the solutions to the given equation are x = x₁, x₂ from the first quadratic equation, and x = -3, x = 1 from the second quadratic equation.