Question

what are the roots of the function f(x)=x² - x + 7? x = (1±27i)/2 x = (1±3√3)/2 x = (1±3i√3)/2 x = ±3i√3

Explanation:

Step1: Recall quadratic - formula

For a quadratic function \(f(x)=ax^{2}+bx + c\), the roots are given by \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). In the function \(f(x)=x^{2}-x + 7\), we have \(a = 1\), \(b=-1\), and \(c = 7\).

Step2: Calculate the discriminant \(\Delta=b^{2}-4ac\)

Substitute \(a = 1\), \(b=-1\), and \(c = 7\) into the discriminant formula. \(\Delta=(-1)^{2}-4\times1\times7=1 - 28=-27\).

Step3: Find the roots using the quadratic - formula

Since \(\Delta=-27\), then \(x=\frac{-(-1)\pm\sqrt{-27}}{2\times1}=\frac{1\pm\sqrt{27}i}{2}=\frac{1\pm3\sqrt{3}i}{2}\).

Answer:

\(x=\frac{1\pm3i\sqrt{3}}{2}\)