Question

612x² - 22x + 6 = 0

Explanation:

Step1: Identify the quadratic equation

The given equation is \(612x^{2}-22x + 6=0\). For a quadratic equation \(ax^{2}+bx + c = 0\), we can use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) or try to simplify the equation first. Let's first simplify the equation by dividing all terms by 2:
\(306x^{2}-11x + 3=0\)

Step2: Calculate the discriminant \(\Delta\)

For the quadratic equation \(ax^{2}+bx + c = 0\), the discriminant \(\Delta=b^{2}-4ac\). Here, \(a = 306\), \(b=- 11\), \(c = 3\).
\(\Delta=(-11)^{2}-4\times306\times3=121-3672=121 - 3672=- 3551\)
Since the discriminant \(\Delta=-3551<0\), the quadratic equation has no real roots. If we consider complex roots, we can use the quadratic formula.
Using the quadratic formula \(x=\frac{-b\pm\sqrt{\Delta}}{2a}\), substituting \(a = 306\), \(b=-11\), \(\Delta=-3551\):
\(x=\frac{11\pm\sqrt{- 3551}}{2\times306}=\frac{11\pm i\sqrt{3551}}{612}\) (where \(i=\sqrt{-1}\) is the imaginary unit)

Answer:

The roots of the equation \(612x^{2}-22x + 6 = 0\) are \(x=\frac{11 + i\sqrt{3551}}{612}\) and \(x=\frac{11 - i\sqrt{3551}}{612}\) (no real roots, complex roots are as above)