7236+4226*x+316*x^2=1000000

asked by guest
on Nov 10, 2024 at 9:29 am



You asked:

Solve the equation \(7236 + 4226 x + 316 \cdot {x}^{2} = 1000000\) for the variable \(x\).

MathBot Answer:

\[\begin{aligned}x &= - \frac{2113}{316} + \frac{\sqrt{318178193}}{316} \approx 49.761234\\x &= - \frac{2113}{316} - \frac{\sqrt{318178193}}{316} \approx -63.134652\end{aligned}\]

Solve by quadratic formula:

Subtract the right hand side from the left hand side of: \[7236 + 4226 x + 316 \cdot {x}^{2} = 1000000\] The result is a quadratic equation: \[316 x^{2} + 4226 x - 992764 = 0\]

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 \(a=316\), \(b=4226\), and \(c=-992764\).

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:\[b^{2}-4ac = 4226^{2} - 1264 \left(-992764\right)=1272712772 > 0\] The discriminant is greater than zero, so this quadratic equation has two real solutions.

The two solutions are: \[x = \frac{\left(-1\right) 4226 + \sqrt{1272712772}}{2 \cdot 316} = - \frac{2113}{316} + \frac{\sqrt{318178193}}{316} \approx 49.761234\] \[x = \frac{\left(-1\right) 4226 - \sqrt{1272712772}}{2 \cdot 316} = - \frac{2113}{316} - \frac{\sqrt{318178193}}{316} \approx -63.134652\]

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