Question

Find all the real solutions of the equation. \(x^3-10 x^2+19 x+30=0\)

Short answer

The real solutions to the equation \(x^3-10x^2+19x+30=0\) are \(x=2\), \(x=3\), and \(x=5\).

Step-by-step solution

Factor the Equation

In this case, factoring the cubic equation \(x^3-10x^2+19x+30=0\) requires identifying factors of 30 that, when subtracted, equal 19, and when multiplied, equal -30. We obtain \( (x-2)(x-3)(x-5)=0 \).

Use the Zero Factor Property

The zero factor property states that if a product of factors equals zero, then at least one of the factors must be zero. So we set each factor, \(x-2\), \(x-3\), and \(x-5\) equal to zero.

Solve Each Factor for X

Solving these gives three solutions: \(x=2\), \(x=3\), and \(x=5\) which are the real solutions to the cubic equation.