Solve sqrt{x}+3x=5000

$$\sqrt{ x } +3x=5000$$

Solve for x

$$x = \frac{30001 - \sqrt{60001}}{18} \approx 1653.113832472$$

Solution Steps

Subtract 3x from both sides of the equation.

$$\sqrt{x}=5000-3x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(5000-3x\right)^{2}$$

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

$$x=\left(5000-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5000-3x\right)^{2}.

$$x=25000000-30000x+9x^{2}$$

Subtract 25000000 from both sides.

$$x-25000000=-30000x+9x^{2}$$

Add 30000x to both sides.

$$x-25000000+30000x=9x^{2}$$

Combine x and 30000x to get 30001x.

$$30001x-25000000=9x^{2}$$

Subtract 9x^{2} from both sides.

$$30001x-25000000-9x^{2}=0$$

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.

$$-9x^{2}+30001x-25000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 30001 for b, and -25000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30001±\sqrt{30001^{2}-4\left(-9\right)\left(-25000000\right)}}{2\left(-9\right)}$$

Square 30001.

$$x=\frac{-30001±\sqrt{900060001-4\left(-9\right)\left(-25000000\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-30001±\sqrt{900060001+36\left(-25000000\right)}}{2\left(-9\right)}$$

Multiply 36 times -25000000.

$$x=\frac{-30001±\sqrt{900060001-900000000}}{2\left(-9\right)}$$

Add 900060001 to -900000000.

$$x=\frac{-30001±\sqrt{60001}}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-30001±\sqrt{60001}}{-18}$$

Now solve the equation x=\frac{-30001±\sqrt{60001}}{-18} when ± is plus. Add -30001 to \sqrt{60001}.

$$x=\frac{\sqrt{60001}-30001}{-18}$$

Divide -30001+\sqrt{60001} by -18.

$$x=\frac{30001-\sqrt{60001}}{18}$$

Now solve the equation x=\frac{-30001±\sqrt{60001}}{-18} when ± is minus. Subtract \sqrt{60001} from -30001.

$$x=\frac{-\sqrt{60001}-30001}{-18}$$

Divide -30001-\sqrt{60001} by -18.

$$x=\frac{\sqrt{60001}+30001}{18}$$

The equation is now solved.

$$x=\frac{30001-\sqrt{60001}}{18} x=\frac{\sqrt{60001}+30001}{18}$$

Substitute \frac{30001-\sqrt{60001}}{18} for x in the equation \sqrt{x}+3x=5000.

$$\sqrt{\frac{30001-\sqrt{60001}}{18}}+3\times \frac{30001-\sqrt{60001}}{18}=5000$$

Simplify. The value x=\frac{30001-\sqrt{60001}}{18} satisfies the equation.

$$5000=5000$$

Substitute \frac{\sqrt{60001}+30001}{18} for x in the equation \sqrt{x}+3x=5000.

$$\sqrt{\frac{\sqrt{60001}+30001}{18}}+3\times \frac{\sqrt{60001}+30001}{18}=5000$$

Simplify. The value x=\frac{\sqrt{60001}+30001}{18} does not satisfy the equation.

$$\frac{15001}{3}+\frac{1}{3}\times 60001^{\frac{1}{2}}=5000$$

Equation \sqrt{x}=5000-3x has a unique solution.

$$x=\frac{30001-\sqrt{60001}}{18}$$