Solve +54x+sqrt{x}=75

$$+54x+ \sqrt{ x } = 75$$

Solve for x

$$x = \frac{8101 - \sqrt{16201}}{5832} \approx 1.3672354$$

Solution Steps

Subtract 54x from both sides of the equation.

$$\sqrt{x}=75-54x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(75-54x\right)^{2}$$

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

$$x=\left(75-54x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(75-54x\right)^{2}.

$$x=5625-8100x+2916x^{2}$$

Subtract 5625 from both sides.

$$x-5625=-8100x+2916x^{2}$$

Add 8100x to both sides.

$$x-5625+8100x=2916x^{2}$$

Combine x and 8100x to get 8101x.

$$8101x-5625=2916x^{2}$$

Subtract 2916x^{2} from both sides.

$$8101x-5625-2916x^{2}=0$$

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.

$$-2916x^{2}+8101x-5625=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2916 for a, 8101 for b, and -5625 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8101±\sqrt{8101^{2}-4\left(-2916\right)\left(-5625\right)}}{2\left(-2916\right)}$$

Square 8101.

$$x=\frac{-8101±\sqrt{65626201-4\left(-2916\right)\left(-5625\right)}}{2\left(-2916\right)}$$

Multiply -4 times -2916.

$$x=\frac{-8101±\sqrt{65626201+11664\left(-5625\right)}}{2\left(-2916\right)}$$

Multiply 11664 times -5625.

$$x=\frac{-8101±\sqrt{65626201-65610000}}{2\left(-2916\right)}$$

Add 65626201 to -65610000.

$$x=\frac{-8101±\sqrt{16201}}{2\left(-2916\right)}$$

Multiply 2 times -2916.

$$x=\frac{-8101±\sqrt{16201}}{-5832}$$

Now solve the equation x=\frac{-8101±\sqrt{16201}}{-5832} when ± is plus. Add -8101 to \sqrt{16201}.

$$x=\frac{\sqrt{16201}-8101}{-5832}$$

Divide -8101+\sqrt{16201} by -5832.

$$x=\frac{8101-\sqrt{16201}}{5832}$$

Now solve the equation x=\frac{-8101±\sqrt{16201}}{-5832} when ± is minus. Subtract \sqrt{16201} from -8101.

$$x=\frac{-\sqrt{16201}-8101}{-5832}$$

Divide -8101-\sqrt{16201} by -5832.

$$x=\frac{\sqrt{16201}+8101}{5832}$$

The equation is now solved.

$$x=\frac{8101-\sqrt{16201}}{5832} x=\frac{\sqrt{16201}+8101}{5832}$$

Substitute \frac{8101-\sqrt{16201}}{5832} for x in the equation 54x+\sqrt{x}=75.

$$54\times \frac{8101-\sqrt{16201}}{5832}+\sqrt{\frac{8101-\sqrt{16201}}{5832}}=75$$

Simplify. The value x=\frac{8101-\sqrt{16201}}{5832} satisfies the equation.

$$75=75$$

Substitute \frac{\sqrt{16201}+8101}{5832} for x in the equation 54x+\sqrt{x}=75.

$$54\times \frac{\sqrt{16201}+8101}{5832}+\sqrt{\frac{\sqrt{16201}+8101}{5832}}=75$$

Simplify. The value x=\frac{\sqrt{16201}+8101}{5832} does not satisfy the equation.

$$\frac{1}{54}\times 16201^{\frac{1}{2}}+\frac{4051}{54}=75$$

Equation \sqrt{x}=75-54x has a unique solution.

$$x=\frac{8101-\sqrt{16201}}{5832}$$