Question: User: 5x² - 8x + 1 = y
position_hint: centerFind the roots
1. **State the problem:** Find the roots of the quadratic equation $$5x^2 - 8x + 1 = 0$$.
2. **Formula used:** The roots of a quadratic equation $$ax^2 + bx + c = 0$$ are given by the quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
3. **Identify coefficients:** Here, $$a = 5$$, $$b = -8$$, and $$c = 1$$.
4. **Calculate the discriminant:**
$$\Delta = b^2 - 4ac = (-8)^2 - 4 \times 5 \times 1 = 64 - 20 = 44$$
5. **Apply the quadratic formula:**
$$x = \frac{-(-8) \pm \sqrt{44}}{2 \times 5} = \frac{8 \pm \sqrt{44}}{10}$$
6. **Simplify the square root:**
$$\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}$$
7. **Substitute back:**
$$x = \frac{8 \pm 2\sqrt{11}}{10}$$
8. **Simplify the fraction by dividing numerator and denominator by 2:**
$$x = \frac{\cancel{2} \times 4 \pm \cancel{2} \times \sqrt{11}}{\cancel{2} \times 5} = \frac{4 \pm \sqrt{11}}{5}$$
9. **Final roots:**
$$x_1 = \frac{4 + \sqrt{11}}{5}, \quad x_2 = \frac{4 - \sqrt{11}}{5}$$
These are the two roots of the quadratic equation.