Question

one of the solutions to the equation 2x^3 - 4x^2 - 30x = 0 is 0. find the full solution set. which of the following statements correctly describes the solution set? (1 point) the equation has all real solutions. the equation has two real solutions and one non - real solution. the equation has one real solution and two non - real solutions. the equation has all non - real solutions.

Explanation:

Step1: Factor out the common factor

Given the equation $2x^{3}-4x^{2}-30x = 0$, factor out $2x$ first. We get $2x(x^{2}-2x - 15)=0$.

Step2: Factor the quadratic expression

Factor $x^{2}-2x - 15$. We need two numbers that multiply to - 15 and add up to - 2. The numbers are - 5 and 3. So $x^{2}-2x - 15=(x - 5)(x+3)$. Then the factored - form of the original equation is $2x(x - 5)(x + 3)=0$.

Step3: Use the zero - product property

Set each factor equal to zero:
If $2x=0$, then $x = 0$.
If $x - 5=0$, then $x=5$.
If $x + 3=0$, then $x=-3$.

Answer:

The equation has all real solutions, and the solution set is $\{-3,0,5\}$.