$$ 2 x^{2}=-x+5 $$ (Are the roots real or Unreal)

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To determine whether the quadratic equation $$ 2x^{2} = -x + 5 $$ has real or unreal (complex) roots, follow these steps:

1. **Rewrite the Equation in Standard Form:**
   
   Move all terms to one side to get the standard quadratic form $$ ax^2 + bx + c = 0 $$:
   $$
   2x^{2} + x - 5 = 0
   $$
   Here, $$ a = 2 $$, $$ b = 1 $$, and $$ c = -5 $$.

2. **Calculate the Discriminant ($$ D $$):**
   
   The discriminant determines the nature of the roots and is given by:
   $$
   D = b^{2} - 4ac
   $$
   Substitute the values:
   $$
   D = (1)^{2} - 4(2)(-5) = 1 + 40 = 41
   $$

3. **Analyze the Discriminant:**
   
   - If $$ D > 0 $$: Two distinct real roots.
   - If $$ D = 0 $$: One real repeated root.
   - If $$ D < 0 $$: Two complex (unreal) roots.
   
   Since $$ D = 41 > 0 $$, the equation has **two distinct real roots**.

4. **(Optional) Find the Roots:**
   
   Using the quadratic formula:
   $$
   x = \frac{-b \pm \sqrt{D}}{2a}
   $$
   $$
   x = \frac{-1 \pm \sqrt{41}}{4}
   $$
   
   This results in two real roots:
   $$
   x = \frac{-1 + \sqrt{41}}{4} \quad \text{and} \quad x = \frac{-1 - \sqrt{41}}{4}
   $$

**Conclusion:** The equation $$ 2x^{2} = -x + 5 $$ has two distinct real roots.
The equation $$ 2x^{2} = -x + 5 $$ has two distinct real roots.

(Are the roots real or Unreal)

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