Solve (x)=3sqrt{x}-4

$$( x ) = 3 \sqrt { x } - 4$$

Solve for x (complex solution)

$$x=\frac{1+3\sqrt{7}i}{2}\approx 0.5+3.968626967i$$

Solution Steps

Subtract -4 from both sides of the equation.

$$x+4=3\sqrt{x}$$

Square both sides of the equation.

$$\left(x+4\right)^{2}=\left(3\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$x^{2}+8x+16=\left(3\sqrt{x}\right)^{2}$$

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

$$x^{2}+8x+16=3^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$x^{2}+8x+16=9\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}+8x+16=9x$$

Subtract 9x from both sides.

$$x^{2}+8x+16-9x=0$$

Combine 8x and -9x to get -x.

$$x^{2}-x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 16}}{2}$$

Multiply -4 times 16.

$$x=\frac{-\left(-1\right)±\sqrt{1-64}}{2}$$

Add 1 to -64.

$$x=\frac{-\left(-1\right)±\sqrt{-63}}{2}$$

Take the square root of -63.

$$x=\frac{-\left(-1\right)±3\sqrt{7}i}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±3\sqrt{7}i}{2}$$

Now solve the equation x=\frac{1±3\sqrt{7}i}{2} when ± is plus. Add 1 to 3i\sqrt{7}.

$$x=\frac{1+3\sqrt{7}i}{2}$$

Now solve the equation x=\frac{1±3\sqrt{7}i}{2} when ± is minus. Subtract 3i\sqrt{7} from 1.

$$x=\frac{-3\sqrt{7}i+1}{2}$$

The equation is now solved.

$$x=\frac{1+3\sqrt{7}i}{2} x=\frac{-3\sqrt{7}i+1}{2}$$

Substitute \frac{1+3\sqrt{7}i}{2} for x in the equation x=3\sqrt{x}-4.

$$\frac{1+3\sqrt{7}i}{2}=3\sqrt{\frac{1+3\sqrt{7}i}{2}}-4$$

Simplify. The value x=\frac{1+3\sqrt{7}i}{2} satisfies the equation.

$$\frac{1}{2}+\frac{3}{2}i\times 7^{\frac{1}{2}}=\frac{1}{2}+\frac{3}{2}i\times 7^{\frac{1}{2}}$$

Substitute \frac{-3\sqrt{7}i+1}{2} for x in the equation x=3\sqrt{x}-4.

$$\frac{-3\sqrt{7}i+1}{2}=3\sqrt{\frac{-3\sqrt{7}i+1}{2}}-4$$

Simplify. The value x=\frac{-3\sqrt{7}i+1}{2} does not satisfy the equation.

$$-\frac{3}{2}i\times 7^{\frac{1}{2}}+\frac{1}{2}=-\frac{17}{2}+\frac{3}{2}i\times 7^{\frac{1}{2}}$$

Equation x+4=3\sqrt{x} has a unique solution.

$$x=\frac{1+3\sqrt{7}i}{2}$$