To solve the equation ( 3(x-2)(x+4) = 2x + x ), we first simplify the right side of the equation:
[
2x + x = 3x
]
Now we can rewrite the equation:
[
3(x-2)(x+4) = 3x
]
Next, we can divide both sides of the equation by 3 (assuming ( x \neq 0 )):
[
(x-2)(x+4) = x
]
Now, let's expand and simplify the left side:
[
x^2 + 4x - 2x - 8 = x
]
This simplifies to:
[
x^2 + 2x - 8 = x
]
Next, we will move all terms to one side of the equation:
[
x^2 + 2x - x - 8 = 0
]
This simplifies to:
[
x^2 + x - 8 = 0
]
Now we can use the quadratic formula ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = 1 ), and ( c = -8 ):
- Calculate the discriminant:
[
b^2 - 4ac = 1^2 - 4(1)(-8) = 1 + 32 = 33
]
- Now, apply the quadratic formula:
[
x = \frac{-1 \pm \sqrt{33}}{2(1)} = \frac{-1 \pm \sqrt{33}}{2}
]
Thus, the solutions for ( x ) are:
[
x = \frac{-1 + \sqrt{33}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{33}}{2}
]
These are the solutions to the equation ( 3(x-2)(x+4) = 2x + x ).
