Quadratic Equation 904205

1. **State the problem:** Solve the quadratic equation $$2n^2 - 15n + 7 = 0$$. 2. **Formula used:** The quadratic formula is $$n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 2$, $b = -15$, and $c = 7$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-15)^2 - 4 \times 2 \times 7 = 225 - 56 = 169$$. 5. **Calculate the roots:** $$n = \frac{-(-15) \pm \sqrt{169}}{2 \times 2} = \frac{15 \pm 13}{4}$$. 6. **Find each solution:** - For the plus sign: $$n = \frac{15 + 13}{4} = \frac{28}{4} = 7$$. - For the minus sign: $$n = \frac{15 - 13}{4} = \frac{2}{4} = \frac{1}{2}$$. 7. **Final answer:** The solutions to the equation are $$n = 7$$ and $$n = \frac{1}{2}$$.