Use the method of your choice to find all real solutions to the equation below. $$ x^{2}=\frac{6}{11} x+\frac{55}{121} $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
x^{2} = \frac{6}{11}x + \frac{55}{121}
$$

follow these steps:

1. **Rearrange the Equation:**

Bring all terms to one side to form a standard quadratic equation:

$$
   x^2 - \frac{6}{11}x - \frac{55}{121} = 0
   $$

2. **Identify Coefficients:**

The quadratic equation is in the form $$ ax^2 + bx + c = 0 $$, where:

- $$ a = 1 $$
   - $$ b = -\frac{6}{11} $$
   - $$ c = -\frac{55}{121} $$

3. **Calculate the Discriminant:**

The discriminant $$ D $$ is given by:

$$
   D = b^2 - 4ac
   $$

Plugging in the values:

$$
   D = \left(-\frac{6}{11}\right)^2 - 4 \cdot 1 \cdot \left(-\frac{55}{121}\right) = \frac{36}{121} + \frac{220}{121} = \frac{256}{121}
   $$

$$
   \sqrt{D} = \frac{16}{11}
   $$

4. **Apply the Quadratic Formula:**

The solutions are:

$$
   x = \frac{-b \pm \sqrt{D}}{2a} = \frac{\frac{6}{11} \pm \frac{16}{11}}{2}
   $$

- **First Solution:**

$$
     x = \frac{\frac{6}{11} + \frac{16}{11}}{2} = \frac{\frac{22}{11}}{2} = \frac{2}{2} = 1
     $$

- **Second Solution:**

$$
     x = \frac{\frac{6}{11} - \frac{16}{11}}{2} = \frac{-\frac{10}{11}}{2} = -\frac{5}{11}
     $$

**Final Answers:**

$$
x = 1 \quad \text{and} \quad x = -\frac{5}{11}
$$

Answer: All real solutions are x = 1 and x = –5⁄11.
The real solutions are $$ x = 1 $$ and $$ x = -\frac{5}{11} $$.

Use the method of your choice to find all real solutions to the equation below. \[ x^{2}=\frac{6}{11} x+\frac{55}{121} \]