2x(x+2) =8x+3

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.