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Answer
The equation \( 2x^{2} = -x + 5 \) has two distinct real roots.
Solution
It appears there might be a small typo in your equation. Assuming the equation is:
\[ 2x^{2} = -x + 5 \]
Let's analyze the roots of the quadratic equation.
### Step 1: Rewrite the Equation in Standard Form
First, bring all terms to one side to express the equation in the standard quadratic form \( ax^2 + bx + c = 0 \):
\[ 2x^2 + x - 5 = 0 \]
### Step 2: Calculate the Discriminant
The discriminant (\( D \)) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting the values from the equation:
- \( a = 2 \)
- \( b = 1 \)
- \( c = -5 \)
\[ D = (1)^2 - 4(2)(-5) \]
\[ D = 1 + 40 \]
\[ D = 41 \]
### Step 3: Determine the Nature of the Roots
- **If \( D > 0 \)**: There are two distinct real roots.
- **If \( D = 0 \)**: There is exactly one real root (a repeated root).
- **If \( D < 0 \)**: The roots are complex (unreal).
Since \( D = 41 > 0 \), the quadratic equation has **two distinct real roots**.
### Conclusion
The roots of the equation \( 2x^{2} = -x + 5 \) are **real and distinct**.
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Simplify this solution Beyond the Answer
To determine if the roots are real or unreal, we start by rearranging the equation into standard form: \( 2x^{2} + x - 5 = 0 \). We can find the discriminant \( D = b^{2} - 4ac \), where \( a = 2 \), \( b = 1 \), and \( c = -5 \). Plugging these values in gives \( D = 1^{2} - 4(2)(-5) = 1 + 40 = 41 \). Since the discriminant is positive, the roots are real! Now, if you ever find yourself in this situation again, remember the discriminant is your best friend! A positive value indicates two distinct real roots, zero means one double root, and a negative value means no real roots, just complex ones. Keep your math toolkit handy, and you'll breeze through quadratic equations!
