Solve 7.5=frac{sqrt{x}}{4}+x/1100

$$7.5= \frac{ \sqrt{ x } }{ 4 } + \frac{ x }{ 1100 }$$

Solve for x

$$x = \frac{92125 - 1375 \sqrt{4345}}{2} \approx 744.827881388$$

Solution Steps

Subtract \frac{x}{1100} from both sides of the equation.

$$7.5-\frac{x}{1100}=\frac{\sqrt{x}}{4}$$

Multiply both sides of the equation by 1100, the least common multiple of 1100,4.

$$8250-x=275\sqrt{x}$$

Square both sides of the equation.

$$\left(8250-x\right)^{2}=\left(275\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8250-x\right)^{2}.

$$68062500-16500x+x^{2}=\left(275\sqrt{x}\right)^{2}$$

Expand \left(275\sqrt{x}\right)^{2}.

$$68062500-16500x+x^{2}=275^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 275 to the power of 2 and get 75625.

$$68062500-16500x+x^{2}=75625\left(\sqrt{x}\right)^{2}$$

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

$$68062500-16500x+x^{2}=75625x$$

Subtract 75625x from both sides.

$$68062500-16500x+x^{2}-75625x=0$$

Combine -16500x and -75625x to get -92125x.

$$68062500-92125x+x^{2}=0$$

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.

$$x^{2}-92125x+68062500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -92125 for b, and 68062500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-92125\right)±\sqrt{\left(-92125\right)^{2}-4\times 68062500}}{2}$$

Square -92125.

$$x=\frac{-\left(-92125\right)±\sqrt{8487015625-4\times 68062500}}{2}$$

Multiply -4 times 68062500.

$$x=\frac{-\left(-92125\right)±\sqrt{8487015625-272250000}}{2}$$

Add 8487015625 to -272250000.

$$x=\frac{-\left(-92125\right)±\sqrt{8214765625}}{2}$$

Take the square root of 8214765625.

$$x=\frac{-\left(-92125\right)±1375\sqrt{4345}}{2}$$

The opposite of -92125 is 92125.

$$x=\frac{92125±1375\sqrt{4345}}{2}$$

Now solve the equation x=\frac{92125±1375\sqrt{4345}}{2} when ± is plus. Add 92125 to 1375\sqrt{4345}.

$$x=\frac{1375\sqrt{4345}+92125}{2}$$

Now solve the equation x=\frac{92125±1375\sqrt{4345}}{2} when ± is minus. Subtract 1375\sqrt{4345} from 92125.

$$x=\frac{92125-1375\sqrt{4345}}{2}$$

The equation is now solved.

$$x=\frac{1375\sqrt{4345}+92125}{2} x=\frac{92125-1375\sqrt{4345}}{2}$$

Substitute \frac{1375\sqrt{4345}+92125}{2} for x in the equation 7.5=\frac{\sqrt{x}}{4}+\frac{x}{1100}.

$$7.5=\frac{\sqrt{\frac{1375\sqrt{4345}+92125}{2}}}{4}+\frac{\frac{1375\sqrt{4345}+92125}{2}}{1100}$$

Simplify. The value x=\frac{1375\sqrt{4345}+92125}{2} does not satisfy the equation.

$$7.5=\frac{305}{4}+\frac{5}{4}\times 4345^{\frac{1}{2}}$$

Substitute \frac{92125-1375\sqrt{4345}}{2} for x in the equation 7.5=\frac{\sqrt{x}}{4}+\frac{x}{1100}.

$$7.5=\frac{\sqrt{\frac{92125-1375\sqrt{4345}}{2}}}{4}+\frac{\frac{92125-1375\sqrt{4345}}{2}}{1100}$$

Simplify. The value x=\frac{92125-1375\sqrt{4345}}{2} satisfies the equation.

$$7.5=\frac{15}{2}$$

Equation 8250-x=275\sqrt{x} has a unique solution.

$$x=\frac{92125-1375\sqrt{4345}}{2}$$