Solve x+3sqrt{x}+4=0

$$x+3 \sqrt{ x } +4 = 0$$

Solve for x (complex solution)

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

Solution Steps

Subtract x+4 from both sides of the equation.

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

To find the opposite of x+4, find the opposite of each term.

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

Square both sides of the equation.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

$$9x=\left(-x-4\right)^{2}$$

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

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

Subtract x^{2} from both sides.

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

Subtract 8x from both sides.

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

Combine 9x and -8x to get x.

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

Subtract 16 from both sides.

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

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.

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

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}.

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

Square 1.

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

Multiply -4 times -1.

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

Multiply 4 times -16.

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

Add 1 to -64.

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

Take the square root of -63.

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

Multiply 2 times -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}.

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

Divide -1+3i\sqrt{7} by -2.

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

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

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

Divide -1-3i\sqrt{7} by -2.

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

The equation is now solved.

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

Substitute \frac{-3\sqrt{7}i+1}{2} for x in the equation x+3\sqrt{x}+4=0.

$$\frac{-3\sqrt{7}i+1}{2}+3\sqrt{\frac{-3\sqrt{7}i+1}{2}}+4=0$$

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

$$0=0$$

Substitute \frac{1+3\sqrt{7}i}{2} for x in the equation x+3\sqrt{x}+4=0.

$$\frac{1+3\sqrt{7}i}{2}+3\sqrt{\frac{1+3\sqrt{7}i}{2}}+4=0$$

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

$$9+3i\times 7^{\frac{1}{2}}=0$$

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

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