To determine the number of real solutions for the equation \( 14f + 49 = -f^2 \), we first rewrite it in standard quadratic form:
- Rearrange the equation: \[ f^2 + 14f + 49 = 0. \]
Next, we identify the coefficients for the quadratic equation \( af^2 + bf + c = 0 \), where:
- \( a = 1 \)
- \( b = 14 \)
- \( c = 49 \)
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To find the number of real solutions, we calculate the discriminant \( D \) using the formula: \[ D = b^2 - 4ac. \]
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Substitute the values for \( a \), \( b \), and \( c \): \[ D = 14^2 - 4(1)(49) = 196 - 196 = 0. \]
The discriminant \( D = 0 \) indicates that there is exactly one real solution to the equation.
Thus, the answer is:
One real solution.