What are the roots of the equation? \[ 15 x^{2}-x-2=-8 x \] Answer \( \frac{-7 \pm i \sqrt{71}}{30} \) \( \frac{2}{3} \) and \( -\frac{1}{5} \) \( \frac{7 \pm i \sqrt{71}}{30} \)

To solve for the roots of the equation \( 15 x^{2}-x-2=-8 x \), we first rearrange the equation into standard form by adding \( 8x \) to both sides. This gives us \( 15x^{2} + 7x - 2 = 0 \). Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 15 \), \( b = 7 \), and \( c = -2 \), we can find the roots. The discriminant \( b^2 - 4ac = 7^2 - 4(15)(-2) = 49 + 120 = 169 \) is a perfect square, leading to rational roots. Confirming this, we get \( x = \frac{-b \pm \sqrt{169}}{2a} = \frac{-7 \pm 13}{30} \), resulting in \( x = \frac{2}{3} \) and \( x = -\frac{1}{5} \). So the correct responses from your options are \( \frac{2}{3} \) and \( -\frac{1}{5} \). Now if you're looking to spice up your problem-solving skills, remember that knowing how to manipulate equations is crucial! Transforming the equation into a standard form is like setting a stage before the big performance—the players of \( a \), \( b \), and \( c \) all need their proper spots to shine! If you want to dive deeper into quadratic equations, check out classic texts like "Algebra" by Harold R. Jacobs or "Algebra and Trigonometry" by Michael Sullivan. These reads not only cover the fundamentals but also provide intriguing problem sets and clear explanations that bring algebra to life!