To solve the assertion and reason question regarding the quadratic equation \( x^2 + 2|x| + 3 = 0 \), we will analyze the assertion and reason step by step.
### Step 1: Understand the Assertion
The assertion states that the equation \( x^2 + 2|x| + 3 = 0 \) has no real roots.
### Step 2: Analyze the Equation
The equation involves an absolute value, which means we need to consider two cases based on the definition of the absolute value.
1. **Case 1:** When \( x \geq 0 \), \( |x| = x \).
- The equation becomes:
\[
x^2 + 2x + 3 = 0
\]
2. **Case 2:** When \( x < 0 \), \( |x| = -x \).
- The equation becomes:
\[
x^2 - 2x + 3 = 0
\]
### Step 3: Calculate the Discriminant for Each Case
To determine if there are real roots, we will calculate the discriminant \( D \) for both cases.
1. **For Case 1:** \( x^2 + 2x + 3 = 0 \)
- Here, \( a = 1, b = 2, c = 3 \).
- The discriminant \( D \) is given by:
\[
D = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]
- Since \( D < 0 \), there are no real roots for this case.
2. **For Case 2:** \( x^2 - 2x + 3 = 0 \)
- Here, \( a = 1, b = -2, c = 3 \).
- The discriminant \( D \) is given by:
\[
D = (-2)^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8
\]
- Again, since \( D < 0 \), there are no real roots for this case as well.
### Step 4: Conclusion on the Assertion
Since both cases yield a negative discriminant, we conclude that the assertion is true: the equation \( x^2 + 2|x| + 3 = 0 \) has no real roots.
### Step 5: Analyze the Reason
The reason states that in a quadratic equation \( ax^2 + bx + c = 0 \), if the discriminant is less than zero, then the equation has no real roots. This is a true statement and correctly explains why the assertion holds.
### Final Conclusion
Both the assertion and the reason are true, and the reason correctly explains the assertion. Therefore, the correct option is:
**Option A:** Assertion and Reason both are true, and Reason explains Assertion.
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