1. **State the problem:** Solve the quadratic equation $y = 2x^2 + 7x - 15$ for $x$ when $y=0$. 2. **Formula used:** To find the roots of a quadratic equation $ax^2 + bx + c = 0$, use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=2$, $b=7$, and $c=-15$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 7^2 - 4 \times 2 \times (-15) = 49 + 120 = 169$$ 4. **Find the square root of the discriminant:** $$\sqrt{\Delta} = \sqrt{169} = 13$$ 5. **Apply the quadratic formula:** $$x = \frac{-7 \pm 13}{2 \times 2} = \frac{-7 \pm 13}{4}$$ 6. **Calculate the two roots:** - For the plus sign: $$x = \frac{-7 + 13}{4} = \frac{6}{4} = \frac{\cancel{6}}{\cancel{4}} = \frac{3}{2} = 1.5$$ - For the minus sign: $$x = \frac{-7 - 13}{4} = \frac{-20}{4} = \frac{\cancel{-20}}{\cancel{4}} = -5$$ 7. **Final answer:** The solutions to the equation $2x^2 + 7x - 15 = 0$ are $$x = 1.5 \quad \text{and} \quad x = -5$$