Math Problem Solver

\[\begin{aligned}x &= \frac{41491}{469373940} + \frac{\sqrt{82882}}{46937394} \approx 9.453001 \cdot 10^{-5}\\x &= \frac{41491}{469373940} - \frac{\sqrt{82882}}{46937394} \approx 8.2262932 \cdot 10^{-5}\end{aligned}\]

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Solve by quadratic formula:

Subtract the right hand side from the left hand side of: \[{\left( 5 - x \cdot 56700 \right)}^{2} = \frac{x}{0.73 \cdot {10}^{-3}}\] The result is a quadratic equation: \[3214890000 x^{2} - \frac{41491000 x}{73} + 25 = 0\]

Given a quadratic equation \(a x^{2} + b x + c = 0\), where \(a\), \(b\), \(c\) are constants and \(a \ne 0\), the solutions are given by the quadratic formula: \[x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\] In this case \(a=3214890000\), \(b=- \frac{41491000}{73}\), and \(c=25\).

The discriminant is the quantity under the square root sign in the quadratic formula, and its sign determines the number of solutions to the quadratic equation when the coefficients are real. The discriminant is:\[b^{2}-4ac = \left(- \frac{41491000}{73}\right)^{2} - 12859560000 \cdot 25=\frac{8288200000000}{5329} > 0\] The discriminant is greater than zero, so this quadratic equation has two real solutions.

The two solutions are: \[x = \frac{\left(-1\right) \left(- \frac{41491000}{73}\right) + \sqrt{\frac{8288200000000}{5329}}}{2 \cdot 3214890000} = \frac{41491}{469373940} + \frac{\sqrt{82882}}{46937394} \approx 9.453001 \cdot 10^{-5}\] \[x = \frac{\left(-1\right) \left(- \frac{41491000}{73}\right) - \sqrt{\frac{8288200000000}{5329}}}{2 \cdot 3214890000} = \frac{41491}{469373940} - \frac{\sqrt{82882}}{46937394} \approx 8.2262932 \cdot 10^{-5}\]