Solve x+3sqrt{x}=30

$$x+3 \sqrt{ x } = 30$$

Solve for x

$$x = \frac{69 - 3 \sqrt{129}}{2} \approx 17.463274963$$

Solution Steps

Subtract x from both sides of the equation.

$$3\sqrt{x}=30-x$$

Square both sides of the equation.

$$\left(3\sqrt{x}\right)^{2}=\left(30-x\right)^{2}$$

Expand \left(3\sqrt{x}\right)^{2}.

$$3^{2}\left(\sqrt{x}\right)^{2}=\left(30-x\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$9\left(\sqrt{x}\right)^{2}=\left(30-x\right)^{2}$$

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

$$9x=\left(30-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(30-x\right)^{2}.

$$9x=900-60x+x^{2}$$

Subtract 900 from both sides.

$$9x-900=-60x+x^{2}$$

Add 60x to both sides.

$$9x-900+60x=x^{2}$$

Combine 9x and 60x to get 69x.

$$69x-900=x^{2}$$

Subtract x^{2} from both sides.

$$69x-900-x^{2}=0$$

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.

$$-x^{2}+69x-900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 69 for b, and -900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-69±\sqrt{69^{2}-4\left(-1\right)\left(-900\right)}}{2\left(-1\right)}$$

Square 69.

$$x=\frac{-69±\sqrt{4761-4\left(-1\right)\left(-900\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-69±\sqrt{4761+4\left(-900\right)}}{2\left(-1\right)}$$

Multiply 4 times -900.

$$x=\frac{-69±\sqrt{4761-3600}}{2\left(-1\right)}$$

Add 4761 to -3600.

$$x=\frac{-69±\sqrt{1161}}{2\left(-1\right)}$$

Take the square root of 1161.

$$x=\frac{-69±3\sqrt{129}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-69±3\sqrt{129}}{-2}$$

Now solve the equation x=\frac{-69±3\sqrt{129}}{-2} when ± is plus. Add -69 to 3\sqrt{129}.

$$x=\frac{3\sqrt{129}-69}{-2}$$

Divide -69+3\sqrt{129} by -2.

$$x=\frac{69-3\sqrt{129}}{2}$$

Now solve the equation x=\frac{-69±3\sqrt{129}}{-2} when ± is minus. Subtract 3\sqrt{129} from -69.

$$x=\frac{-3\sqrt{129}-69}{-2}$$

Divide -69-3\sqrt{129} by -2.

$$x=\frac{3\sqrt{129}+69}{2}$$

The equation is now solved.

$$x=\frac{69-3\sqrt{129}}{2} x=\frac{3\sqrt{129}+69}{2}$$

Substitute \frac{69-3\sqrt{129}}{2} for x in the equation x+3\sqrt{x}=30.

$$\frac{69-3\sqrt{129}}{2}+3\sqrt{\frac{69-3\sqrt{129}}{2}}=30$$

Simplify. The value x=\frac{69-3\sqrt{129}}{2} satisfies the equation.

$$30=30$$

Substitute \frac{3\sqrt{129}+69}{2} for x in the equation x+3\sqrt{x}=30.

$$\frac{3\sqrt{129}+69}{2}+3\sqrt{\frac{3\sqrt{129}+69}{2}}=30$$

Simplify. The value x=\frac{3\sqrt{129}+69}{2} does not satisfy the equation.

$$3\times 129^{\frac{1}{2}}+39=30$$

Equation 3\sqrt{x}=30-x has a unique solution.

$$x=\frac{69-3\sqrt{129}}{2}$$