Solve x-3sqrt{x}-2=0

$$x - 3 \sqrt { x } - 2 = 0$$

Solve for x

$$x = \frac{3 \sqrt{17} + 13}{2} \approx 12.684658438$$

Solution Steps

Add 2 to both sides. Anything plus zero gives itself.

$$x-3\sqrt{x}=2$$

Subtract x from both sides of the equation.

$$-3\sqrt{x}=2-x$$

Square both sides of the equation.

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

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

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

Calculate -3 to the power of 2 and get 9.

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

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

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

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

$$9x=4-4x+x^{2}$$

Add 4x to both sides.

$$9x+4x=4+x^{2}$$

Combine 9x and 4x to get 13x.

$$13x=4+x^{2}$$

Subtract x^{2} from both sides.

$$13x-x^{2}=4$$

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.

$$-x^{2}+13x=4$$

Subtract 4 from both sides of the equation.

$$-x^{2}+13x-4=4-4$$

Subtracting 4 from itself leaves 0.

$$-x^{2}+13x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 13 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-13±\sqrt{169+4\left(-4\right)}}{2\left(-1\right)}$$

Multiply 4 times -4.

$$x=\frac{-13±\sqrt{169-16}}{2\left(-1\right)}$$

Add 169 to -16.

$$x=\frac{-13±\sqrt{153}}{2\left(-1\right)}$$

Take the square root of 153.

$$x=\frac{-13±3\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-13±3\sqrt{17}}{-2}$$

Now solve the equation x=\frac{-13±3\sqrt{17}}{-2} when ± is plus. Add -13 to 3\sqrt{17}.

$$x=\frac{3\sqrt{17}-13}{-2}$$

Divide -13+3\sqrt{17} by -2.

$$x=\frac{13-3\sqrt{17}}{2}$$

Now solve the equation x=\frac{-13±3\sqrt{17}}{-2} when ± is minus. Subtract 3\sqrt{17} from -13.

$$x=\frac{-3\sqrt{17}-13}{-2}$$

Divide -13-3\sqrt{17} by -2.

$$x=\frac{3\sqrt{17}+13}{2}$$

The equation is now solved.

$$x=\frac{13-3\sqrt{17}}{2} x=\frac{3\sqrt{17}+13}{2}$$

Substitute \frac{13-3\sqrt{17}}{2} for x in the equation x-3\sqrt{x}-2=0.

$$\frac{13-3\sqrt{17}}{2}-3\sqrt{\frac{13-3\sqrt{17}}{2}}-2=0$$

Simplify. The value x=\frac{13-3\sqrt{17}}{2} does not satisfy the equation.

$$9-3\times 17^{\frac{1}{2}}=0$$

Substitute \frac{3\sqrt{17}+13}{2} for x in the equation x-3\sqrt{x}-2=0.

$$\frac{3\sqrt{17}+13}{2}-3\sqrt{\frac{3\sqrt{17}+13}{2}}-2=0$$

Simplify. The value x=\frac{3\sqrt{17}+13}{2} satisfies the equation.

$$0=0$$

Equation -3\sqrt{x}=2-x has a unique solution.

$$x=\frac{3\sqrt{17}+13}{2}$$