Question

find all complex solutions of $7x^2 + 7x + 2 = 0$. (if there is more than one solution, separate them with commas.)

Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are given by \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 7\), \(b = 7\), \(c = 2\).

Step2: Calculate Discriminant

Discriminant \(D=b^2 - 4ac=(7)^2-4\times7\times2 = 49 - 56=-7\).

Step3: Apply Quadratic Formula

Substitute \(a\), \(b\), \(D\) into the formula:
\(x=\frac{-7\pm\sqrt{-7}}{2\times7}=\frac{-7\pm i\sqrt{7}}{14}\) (since \(\sqrt{-7}=i\sqrt{7}\) for complex numbers).

Answer:

\(\frac{-7 + i\sqrt{7}}{14},\frac{-7 - i\sqrt{7}}{14}\)