X=7:2x-11

To solve the equation \( x = 7:2x - 11 \), we first interpret the expression \( 7:2x \). It typically represents division, so we rewrite it as:

\[ x = \frac{7}{2x} - 11 \]

Now, we can solve for \( x \). First, we'll eliminate the fraction by multiplying both sides by \( 2x \) (assuming \( x \neq 0 \)):

\[ 2x^2 = 7 - 22x \]

Next, rearranging this gives us a standard quadratic equation:

\[ 2x^2 + 22x - 7 = 0 \]

Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2 \), \( b = 22 \), and \( c = -7 \).

Calculating the discriminant:

\[ b^2 - 4ac = 22^2 - 4(2)(-7) = 484 + 56 = 540 \]

So now we find \( x \):

\[ x = \frac{-22 \pm \sqrt{540}}{2(2)} = \frac{-22 \pm \sqrt{540}}{4} \]

The square root of \( 540 \) can be simplified:

\[ \sqrt{540} = \sqrt{4 \times 135} = 2\sqrt{135} \]

Continuing, we express \( \sqrt{135} \):

\[ \sqrt{135} = \sqrt{9 \times 15} = 3\sqrt{15} \]

Thus:

\[ \sqrt{540} = 2(3\sqrt{15}) = 6\sqrt{15} \]

Now we substitute back:

\[ x = \frac{-22 \pm 6\sqrt{15}}{4} \]

This can be simplified further:

\[ x = \frac{-11 \pm 3\sqrt{15}}{2} \]

Thus, the solutions for \( x \) are:

\[ x = \frac{-11 + 3\sqrt{15}}{2} \quad \text{and} \quad x = \frac{-11 - 3\sqrt{15}}{2} \]