Question Use the discriminant to determine the number of solutions to the quadratic equation \( -81 a^{2}+90 a-25=0 \). Provide your answer below:
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To determine the number of solutions to the quadratic equation \(-81 a^{2}+90 a-25=0\), we first calculate the discriminant \(D\) using the formula \(D = b^2 - 4ac\), where \(a = -81\), \(b = 90\), and \(c = -25\). Calculating the discriminant: \[ D = (90)^2 - 4 \cdot (-81) \cdot (-25) \] \[ D = 8100 - 8100 = 0 \] Since the discriminant \(D = 0\), this indicates that the quadratic equation has exactly one real solution (a repeated root). So, the number of solutions to the equation \(-81 a^{2}+90 a-25=0\) is **one solution**.