Find an equation that has the given solutions: $x=\frac{-1 \pm 5 i}{2}$ Write your answer in standard form. Equation: Question Help: Video eBook Message instructor Submit Question

Step 1 ：The given solutions are complex numbers. The solutions of a quadratic equation are given by the formula \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Comparing this with the given solutions, we can see that \(b=1\), \(a=2\) and the discriminant \(b^2-4ac\) is \(-25\). We can use these values to find the coefficients of the quadratic equation.

Step 2 ：Let's denote the solutions as \(x1 = (-0.5+2.5j)\) and \(x2 = (-0.5-2.5j)\).

Step 3 ：From the quadratic formula, we can set up the following system of equations: \(2*a = 1\) (eq1), \(b = -1\) (eq2), and \(-4*a*c + b^2 = -100.0\) (eq3).

Step 4 ：Solving this system of equations gives the coefficients of the quadratic equation. The coefficient \(a\) is \(0.5\), \(b\) is \(-1\), and \(c\) is \(50.5\).

Step 5 ：Therefore, the quadratic equation that has the given solutions is \(0.5x^2 - x + 50.5 = 0\).

Step 6 ：Final Answer: The quadratic equation that has the given solutions is \(\boxed{0.5x^2 - x + 50.5 = 0}\).