Question: 4.x^2 + 5x - 7 = 0
1. **State the problem:** Solve the quadratic equation $$4x^2 + 5x - 7 = 0$$.
2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the equation is in the form $$ax^2 + bx + c = 0$$.
3. **Identify coefficients:** Here, $$a = 4$$, $$b = 5$$, and $$c = -7$$.
4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 4 \times (-7) = 25 + 112 = 137$$.
5. **Apply the quadratic formula:**
$$x = \frac{-5 \pm \sqrt{137}}{2 \times 4} = \frac{-5 \pm \sqrt{137}}{8}$$.
6. **Final answer:** The solutions are $$x = \frac{-5 + \sqrt{137}}{8}$$ and $$x = \frac{-5 - \sqrt{137}}{8}$$.