Solve for all solutions of x : \[ x^{2}+10 x-3=0 \]

To solve the quadratic equation \( x^{2} + 10x - 3 = 0 \), we can use the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our equation, \( a = 1 \), \( b = 10 \), and \( c = -3 \). Substituting these values into the formula gives us: \[ x = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \] Calculating the discriminant: \[ 10^2 - 4 \cdot 1 \cdot (-3) = 100 + 12 = 112 \] Now we can plug this back into the formula: \[ x = \frac{-10 \pm \sqrt{112}}{2} \] Since \( \sqrt{112} = \sqrt{16 \cdot 7} = 4\sqrt{7} \), we have: \[ x = \frac{-10 \pm 4\sqrt{7}}{2} \] Dividing each term by 2 results in the final solution: \[ x = -5 \pm 2\sqrt{7} \] Thus, the solutions are: \[ x = -5 + 2\sqrt{7} \quad \text{and} \quad x = -5 - 2\sqrt{7} \]