Solve sqrt{x-2}=3quadx=

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

Solve for x (complex solution)

$$x=\frac{1+\sqrt{71}i}{18}\approx 0.055555556+0.468119432i$$

Solution Steps

Square both sides of the equation.

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

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

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

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

$$x-2=3^{2}x^{2}$$

Calculate 3 to the power of 2 and get 9.

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

Subtract 9x^{2} from both sides.

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

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.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, 1 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-9\right)\left(-2\right)}}{2\left(-9\right)}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-9\right)\left(-2\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-1±\sqrt{1+36\left(-2\right)}}{2\left(-9\right)}$$

Multiply 36 times -2.

$$x=\frac{-1±\sqrt{1-72}}{2\left(-9\right)}$$

Add 1 to -72.

$$x=\frac{-1±\sqrt{-71}}{2\left(-9\right)}$$

Take the square root of -71.

$$x=\frac{-1±\sqrt{71}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{-1±\sqrt{71}i}{-18}$$

Now solve the equation x=\frac{-1±\sqrt{71}i}{-18} when ± is plus. Add -1 to i\sqrt{71}.

$$x=\frac{-1+\sqrt{71}i}{-18}$$

Divide -1+i\sqrt{71} by -18.

$$x=\frac{-\sqrt{71}i+1}{18}$$

Now solve the equation x=\frac{-1±\sqrt{71}i}{-18} when ± is minus. Subtract i\sqrt{71} from -1.

$$x=\frac{-\sqrt{71}i-1}{-18}$$

Divide -1-i\sqrt{71} by -18.

$$x=\frac{1+\sqrt{71}i}{18}$$

The equation is now solved.

$$x=\frac{-\sqrt{71}i+1}{18} x=\frac{1+\sqrt{71}i}{18}$$

Substitute \frac{-\sqrt{71}i+1}{18} for x in the equation \sqrt{x-2}=3x.

$$\sqrt{\frac{-\sqrt{71}i+1}{18}-2}=3\times \frac{-\sqrt{71}i+1}{18}$$

Simplify. The value x=\frac{-\sqrt{71}i+1}{18} does not satisfy the equation.

$$-\left(\frac{1}{6}-\frac{1}{6}i\times 71^{\frac{1}{2}}\right)=-\frac{1}{6}i\times 71^{\frac{1}{2}}+\frac{1}{6}$$

Substitute \frac{1+\sqrt{71}i}{18} for x in the equation \sqrt{x-2}=3x.

$$\sqrt{\frac{1+\sqrt{71}i}{18}-2}=3\times \frac{1+\sqrt{71}i}{18}$$

Simplify. The value x=\frac{1+\sqrt{71}i}{18} satisfies the equation.

$$\frac{1}{6}+\frac{1}{6}i\times 71^{\frac{1}{2}}=\frac{1}{6}+\frac{1}{6}i\times 71^{\frac{1}{2}}$$

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

$$x=\frac{1+\sqrt{71}i}{18}$$