Solve for the roots in simplest form using the quadratic formula: $$ 4 x^{2}-20 x=-13 $$

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To solve the quadratic equation $$ 4x^{2} - 20x = -13 $$ using the quadratic formula, follow these steps:

1. **Rewrite the equation in standard form:**
   $$
   4x^{2} - 20x + 13 = 0
   $$
   
2. **Identify the coefficients:**
   $$
   a = 4, \quad b = -20, \quad c = 13
   $$
   
3. **Apply the quadratic formula:**
   $$
   x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
   $$
   
4. **Plug in the values of $$ a $$, $$ b $$, and $$ c $$:**
   $$
   x = \frac{20 \pm \sqrt{(-20)^{2} - 4 \cdot 4 \cdot 13}}{2 \cdot 4}
   $$
   $$
   x = \frac{20 \pm \sqrt{400 - 208}}{8}
   $$
   $$
   x = \frac{20 \pm \sqrt{192}}{8}
   $$
   
5. **Simplify the square root:**
   $$
   \sqrt{192} = \sqrt{64 \cdot 3} = 8\sqrt{3}
   $$
   
6. **Substitute back into the equation:**
   $$
   x = \frac{20 \pm 8\sqrt{3}}{8}
   $$
   
7. **Simplify the expression by dividing numerator and denominator by 4:**
   $$
   x = \frac{5 \pm 2\sqrt{3}}{2}
   $$

**Final Answer:**
$$
x = \frac{5 \pm 2\sqrt{3}}{2}
$$
The roots are $$ x = \frac{5 + 2\sqrt{3}}{2} $$ and $$ x = \frac{5 - 2\sqrt{3}}{2} $$.

Solve for the roots in simplest form using the quadratic formula: \[ 4 x^{2}-20 x=-13 \]

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