Solve 485x-9sqrt{4x}=104^2-64x

$$485x-9 \sqrt{ 4x } = { 104 }^{ 2 } -64x$$

Solve for x

$$x=\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\approx 19.847341785$$

Solution Steps

Subtract 485x from both sides of the equation.

$$-9\sqrt{4x}=104^{2}-64x-485x$$

Calculate 104 to the power of 2 and get 10816.

$$-9\sqrt{4x}=10816-64x-485x$$

Combine -64x and -485x to get -549x.

$$-9\sqrt{4x}=10816-549x$$

Square both sides of the equation.

$$\left(-9\sqrt{4x}\right)^{2}=\left(10816-549x\right)^{2}$$

Expand \left(-9\sqrt{4x}\right)^{2}.

$$\left(-9\right)^{2}\left(\sqrt{4x}\right)^{2}=\left(10816-549x\right)^{2}$$

Calculate -9 to the power of 2 and get 81.

$$81\left(\sqrt{4x}\right)^{2}=\left(10816-549x\right)^{2}$$

Calculate \sqrt{4x} to the power of 2 and get 4x.

$$81\times 4x=\left(10816-549x\right)^{2}$$

Multiply 81 and 4 to get 324.

$$324x=\left(10816-549x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10816-549x\right)^{2}.

$$324x=116985856-11875968x+301401x^{2}$$

Subtract 116985856 from both sides.

$$324x-116985856=-11875968x+301401x^{2}$$

Add 11875968x to both sides.

$$324x-116985856+11875968x=301401x^{2}$$

Combine 324x and 11875968x to get 11876292x.

$$11876292x-116985856=301401x^{2}$$

Subtract 301401x^{2} from both sides.

$$11876292x-116985856-301401x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-301401x^{2}[phone]x-116985856=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -301401 for a, 11876292 for b, and -116985856 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11876292±\sqrt{11876292^{2}-4\left(-301401\right)\left(-116985856\right)}}{2\left(-301401\right)}$$

Square 11876292.

$$x=\frac{-11876292±\sqrt{141046311669264-4\left(-301401\right)\left(-116985856\right)}}{2\left(-301401\right)}$$

Multiply -4 times -301401.

$$x=\frac{-11876292±\sqrt{141046311669264+1205604\left(-116985856\right)}}{2\left(-301401\right)}$$

Multiply 1205604 times -116985856.

$$x=\frac{-11876292±\sqrt{141046311669264-141038615937024}}{2\left(-301401\right)}$$

Add 141046311669264 to -141038615937024.

$$x=\frac{-11876292±\sqrt{7695732240}}{2\left(-301401\right)}$$

Take the square root of 7695732240.

$$x=\frac{-11876292±756\sqrt{13465}}{2\left(-301401\right)}$$

Multiply 2 times -301401.

$$x=\frac{-11876292±756\sqrt{13465}}{-602802}$$

Now solve the equation x=\frac{-11876292±756\sqrt{13465}}{-602802} when ± is plus. Add -11876292 to 756\sqrt{13465}.

$$x=\frac{756\sqrt{13465}-11876292}{-602802}$$

Divide -11876292+756\sqrt{13465} by -602802.

$$x=-\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}$$

Now solve the equation x=\frac{-11876292±756\sqrt{13465}}{-602802} when ± is minus. Subtract 756\sqrt{13465} from -11876292.

$$x=\frac{-756\sqrt{13465}-11876292}{-602802}$$

Divide -11876292-756\sqrt{13465} by -602802.

$$x=\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}$$

The equation is now solved.

$$x=-\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489} x=\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}$$

Substitute -\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489} for x in the equation 485x-9\sqrt{4x}=104^{2}-64x.

$$485\left(-\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\right)-9\sqrt{4\left(-\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\right)}=104^{2}-64\left(-\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\right)$$

Simplify. The value x=-\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489} does not satisfy the equation.

$$-\frac{14476}{11163}\times 13465^{\frac{1}{2}}+\frac{320009972}{33489}=\frac{319990208}{33489}+\frac{896}{11163}\times 13465^{\frac{1}{2}}$$

Substitute \frac{14\sqrt{13465}}{11163}+\frac{659794}{33489} for x in the equation 485x-9\sqrt{4x}=104^{2}-64x.

$$485\left(\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\right)-9\sqrt{4\left(\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\right)}=104^{2}-64\left(\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}\right)$$

Simplify. The value x=\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489} satisfies the equation.

$$-\frac{896}{11163}\times 13465^{\frac{1}{2}}+\frac{319990208}{33489}=\frac{319990208}{33489}-\frac{896}{11163}\times 13465^{\frac{1}{2}}$$

Equation -9\sqrt{4x}=10816-549x has a unique solution.

$$x=\frac{14\sqrt{13465}}{11163}+\frac{659794}{33489}$$