Let's open up one of the brackets first
x(3x^2 + 15x - 2x - 10) = 4x + 2
Let's open up the last bracket
3x^3 + 15x^2 - 2x^2 - 10x = 4x + 2
Now let's move all the variables to the left hand side
3x^3 + 13x^2 -14x - 2 = 0
Now we have a cubic equation.
To solve this, we use the factor theorem.
f(x) = 3x^3 + 13x^2 -14x - 2
When f(a) = 0, then (x-a) is a factor of f(x)
f(0) = -2
f(-1) = -3 + 13 + 14 -2 = 22
f(1) = 3 + 13 -14 -2 = 0
f(1) = 0, therefore (x-1) is a factor of f(x)
We now need to divide f(x) by (x-1) and find a quadratic equation.
I have 'magically' divided it. (using the long division method)
f(x) = (3x^2 + 16x + 2)(x-1)
We now have a quadratic and a linear.
(3x^2 + 16x + 2)(x-1) = 0
x-1 = 0
x = 1 --> First solution
3x^2 + 16x + 2 = 0
Quadratic equation (in our case ) has the following solutons:
For these solutions to exist, the discriminant should not be a negative number.
First, we need to compute the discriminant : .
Discriminant d=232 is greater than zero. That means that there are two solutions: .
Quadratic expression can be factored:
Again, the answer is: -0.128075631378697, -5.20525770195464.
Here's your graph: