Question

x² + 4x + 7 = 0

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = 4\), \(c = 7\).

Step2: Calculate discriminant \(\Delta\)

Discriminant formula: \(\Delta = b^2 - 4ac\)
Substitute values: \(\Delta = 4^2 - 4\times1\times7 = 16 - 28 = -12\)

Step3: Solve for \(x\) using quadratic formula \(x=\frac{-b\pm\sqrt{\Delta}}{2a}\)

Since \(\Delta = -12\), \(\sqrt{\Delta}=\sqrt{-12}=2i\sqrt{3}\) (where \(i\) is imaginary unit, \(i^2=-1\))
Substitute into formula: \(x=\frac{-4\pm2i\sqrt{3}}{2\times1}=\frac{-4\pm2i\sqrt{3}}{2}=-2\pm i\sqrt{3}\)

Answer:

The solutions are \(x = -2 + i\sqrt{3}\) and \(x = -2 - i\sqrt{3}\)