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Question Use the discriminant to determine the number of real solutions to the quadratic equation. \[ 81 d^{2}+18 d+1=0 \]
Ask by Chavez Johnson. in the United States
Feb 27,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The quadratic equation has exactly one real solution.
Solution
For the quadratic equation 81d² + 18d + 1 = 0, we identify the coefficients as follows:
a = 81, b = 18, c = 1.
The discriminant Δ is given by:
Δ = b² – 4ac = (18)² – 4·81·1 = 324 – 324 = 0.
Since the discriminant is 0, the quadratic has exactly one real solution (a repeated root).
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Simplify this solution Beyond the Answer
To find the number of real solutions for the quadratic equation \( 81 d^{2}+18 d+1=0 \), we first identify the coefficients: \( a = 81 \), \( b = 18 \), and \( c = 1 \). The discriminant \( D \) is calculated using the formula \( D = b^2 - 4ac \). Plugging in the values: \[ D = 18^2 - 4 \cdot 81 \cdot 1 = 324 - 324 = 0 \] Since the discriminant \( D = 0 \), this indicates that there is exactly one real solution to the quadratic equation.
