1. **Stating the problem:** Solve the quadratic equation $$5x^2 - 4x - 1 = 0$$. 2. **Formula used:** The quadratic formula is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where the quadratic equation is $$ax^2 + bx + c = 0$$. 3. **Identify coefficients:** Here, $$a=5$$, $$b=-4$$, and $$c=-1$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 5 \times (-1) = 16 + 20 = 36$$. 5. **Apply the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{36}}{2 \times 5} = \frac{4 \pm 6}{10}$$. 6. **Find the two solutions:** - For the plus sign: $$x = \frac{4 + 6}{10} = \frac{10}{10} = 1$$. - For the minus sign: $$x = \frac{4 - 6}{10} = \frac{-2}{10} = -\frac{1}{5}$$. --- 1. **Stating the problem:** Solve the quadratic equation $$x^2 + 2x + 7 = 0$$. 2. **Formula used:** Same quadratic formula as above. 3. **Identify coefficients:** $$a=1$$, $$b=2$$, $$c=7$$. 4. **Calculate the discriminant:** $$\Delta = 2^2 - 4 \times 1 \times 7 = 4 - 28 = -24$$. 5. **Since $$\Delta < 0$$, the solutions are complex:** $$x = \frac{-2 \pm \sqrt{-24}}{2} = \frac{-2 \pm \sqrt{24}i}{2} = \frac{-2 \pm 2\sqrt{6}i}{2}$$. 6. **Simplify:** $$x = -1 \pm \sqrt{6}i$$. --- 1. **Stating the problem:** Solve the quadratic equation $$4x^2 - 12x + 9 = 0$$. 2. **Formula used:** Quadratic formula. 3. **Identify coefficients:** $$a=4$$, $$b=-12$$, $$c=9$$. 4. **Calculate the discriminant:** $$\Delta = (-12)^2 - 4 \times 4 \times 9 = 144 - 144 = 0$$. 5. **Since $$\Delta = 0$$, there is one real repeated root:** $$x = \frac{-(-12)}{2 \times 4} = \frac{12}{8} = \frac{3}{2}$$. --- **Final answers:** - For $$5x^2 - 4x - 1 = 0$$: $$x = 1$$ or $$x = -\frac{1}{5}$$. - For $$x^2 + 2x + 7 = 0$$: $$x = -1 + \sqrt{6}i$$ or $$x = -1 - \sqrt{6}i$$. - For $$4x^2 - 12x + 9 = 0$$: $$x = \frac{3}{2}$$ (double root).