To solve the equation (2x(x+2) = 8x + 3), we can start by expanding the left side:
[
2x^2 + 4x = 8x + 3
]
Next, we'll rearrange the equation by moving all terms to one side:
[
2x^2 + 4x - 8x - 3 = 0
]
This simplifies to:
[
2x^2 - 4x - 3 = 0
]
Next, we can either factor this quadratic equation or use the quadratic formula. Let's use the quadratic formula, which is given by:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
In our equation, (a = 2), (b = -4), and (c = -3).
First, we calculate the discriminant (b^2 - 4ac):
[
b^2 - 4ac = (-4)^2 - 4(2)(-3) = 16 + 24 = 40
]
Now, we apply the quadratic formula:
[
x = \frac{-(-4) \pm \sqrt{40}}{2(2)}
]
This simplifies to:
[
x = \frac{4 \pm \sqrt{40}}{4}
]
We can simplify (\sqrt{40}) as follows:
[
\sqrt{40} = \sqrt{4 \cdot 10} = 2\sqrt{10}
]
So, the equation for (x) becomes:
[
x = \frac{4 \pm 2\sqrt{10}}{4}
]
This simplifies to:
[
x = 1 \pm \frac{\sqrt{10}}{2}
]
Thus, the solutions are:
[
x = 1 + \frac{\sqrt{10}}{2} \quad \text{and} \quad x = 1 - \frac{\sqrt{10}}{2}
]
These are the final solutions to the equation.
