$$ 1 \leftarrow \quad $$ Solve the equation. $$ \frac{4}{x^{2}}-\frac{36}{x}=19 $$

Question

UpStudy AI · Accepted Answer

To solve the equation:

$$
\frac{4}{x^{2}} - \frac{36}{x} = 19
$$

**Step 1: Multiply Both Sides by $$ x^{2} $$ to Eliminate Fractions**

$$
4 - 36x = 19x^{2}
$$

**Step 2: Rearrange the Equation into Standard Quadratic Form**

$$
19x^{2} + 36x - 4 = 0
$$

**Step 3: Apply the Quadratic Formula**

For a quadratic equation $$ ax^{2} + bx + c = 0 $$, the solutions are:

$$
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
$$

Here, $$ a = 19 $$, $$ b = 36 $$, and $$ c = -4 $$.

$$
x = \frac{-36 \pm \sqrt{36^{2} - 4 \cdot 19 \cdot (-4)}}{2 \cdot 19}
$$

$$
x = \frac{-36 \pm \sqrt{1296 + 304}}{38}
$$

$$
x = \frac{-36 \pm \sqrt{1600}}{38}
$$

$$
x = \frac{-36 \pm 40}{38}
$$

**Step 4: Calculate the Two Possible Solutions**

1. **First Solution:**

$$
x = \frac{-36 + 40}{38} = \frac{4}{38} = \frac{2}{19}
$$

2. **Second Solution:**

$$
x = \frac{-36 - 40}{38} = \frac{-76}{38} = -2
$$

**Final Solutions:**

$$
x = \frac{2}{19} \quad \text{and} \quad x = -2
$$
The solutions to the equation are $$ x = \frac{2}{19} $$ and $$ x = -2 $$.

\( 1 \leftarrow \quad \) Solve the equation. \[ \frac{4}{x^{2}}-\frac{36}{x}=19 \]