Ask AI: What are the two roots of 9x^2-12x+7=0

To solve the quadratic equation \( 9x^2 - 12x + 7 = 0 \) for its roots, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Given the equation \( 9x^2 - 12x + 7 = 0 \), we have: - \( a = 9 \) - \( b = -12 \) - \( c = 7 \) First, compute the discriminant \( \Delta \): \[ \Delta = b^2 - 4ac = (-12)^2 - 4(9)(7) = 144 - 252 = -108 \] Since the discriminant is negative (\( \Delta = -108 \)), the equation has two complex roots. Next, we compute the roots using the quadratic formula: \[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \] Substitute the values for \( a \), \( b \), and \( \Delta \): \[ x = \frac{-(-12) \pm \sqrt{-108}}{2(9)} = \frac{12 \pm \sqrt{-108}}{18} \] Since \( \sqrt{-108} \) can be expressed as \( \sqrt{-1 \cdot 108} = \sqrt{108}i \), and \( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \), we have \( \sqrt{-108} = 6\sqrt{3}i \). Therefore: \[ x = \frac{12 \pm 6\sqrt{3}i}{18} \] Simplify: \[ x = \frac{12}{18} \pm \frac{6\sqrt{3}i}{18} \] \[ x = \frac{2}{3} \pm \frac{\sqrt{3}i}{3} \] Thus, the two roots of the quadratic equation \( 9x^2 - 12x + 7 = 0 \) are: \[ x = \frac{2}{3} + \frac{\sqrt{3}i}{3} \] \[ x = \frac{2}{3} - \frac{\sqrt{3}i}{3} \] Or equivalently: \[ x = \frac{2}{3} + \frac{\sqrt{3}}{3}i \] \[ x = \frac{2}{3} - \frac{\sqrt{3}}{3}i \]