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If f(x)=x^(2)+x+1 defined on R to R, fin...

If `f(x)=x^(2)+x+1` defined on `R to R`, find value of `x` for which `f(x)=-1`.

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To solve the problem, we need to find the value of \( x \) for which \( f(x) = -1 \), where \( f(x) = x^2 + x + 1 \). ### Step-by-Step Solution: 1. **Set the equation**: We start by equating the function to -1: \[ x^2 + x + 1 = -1 \] 2. **Rearrange the equation**: Move -1 to the left side of the equation: \[ x^2 + x + 1 + 1 = 0 \] This simplifies to: \[ x^2 + x + 2 = 0 \] 3. **Identify coefficients**: In the quadratic equation \( ax^2 + bx + c = 0 \), we have: - \( a = 1 \) - \( b = 1 \) - \( c = 2 \) 4. **Calculate the discriminant**: The discriminant \( D \) is given by the formula: \[ D = b^2 - 4ac \] Plugging in the values: \[ D = 1^2 - 4 \cdot 1 \cdot 2 = 1 - 8 = -7 \] 5. **Analyze the discriminant**: Since the discriminant \( D \) is less than 0 (\( D < 0 \)), it indicates that the quadratic equation has no real roots. This means there are no real values of \( x \) that satisfy the equation \( x^2 + x + 2 = 0 \). 6. **Conclusion**: Therefore, there is no value of \( x \) for which \( f(x) = -1 \). ### Final Answer: No such value of \( x \) exists for which \( f(x) = -1 \).
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