Solve 4x+3/2=sqrt{x}

$$4x+ \frac{ 3 }{ 2 } = \sqrt{ x }$$

Solve for x (complex solution)

$$x=\frac{-11+\sqrt{23}i}{32}\approx -0.34375+0.149869735i$$

Solution Steps

Square both sides of the equation.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+\frac{3}{2}\right)^{2}.

$$16x^{2}+12x+\frac{9}{4}=\left(\sqrt{x}\right)^{2}$$

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

$$16x^{2}+12x+\frac{9}{4}=x$$

Subtract x from both sides.

$$16x^{2}+12x+\frac{9}{4}-x=0$$

Combine 12x and -x to get 11x.

$$16x^{2}+11x+\frac{9}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 11 for b, and \frac{9}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 16\times \frac{9}{4}}}{2\times 16}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 16\times \frac{9}{4}}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-11±\sqrt{121-64\times \frac{9}{4}}}{2\times 16}$$

Multiply -64 times \frac{9}{4}.

$$x=\frac{-11±\sqrt{121-144}}{2\times 16}$$

Add 121 to -144.

$$x=\frac{-11±\sqrt{-23}}{2\times 16}$$

Take the square root of -23.

$$x=\frac{-11±\sqrt{23}i}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-11±\sqrt{23}i}{32}$$

Now solve the equation x=\frac{-11±\sqrt{23}i}{32} when ± is plus. Add -11 to i\sqrt{23}.

$$x=\frac{-11+\sqrt{23}i}{32}$$

Now solve the equation x=\frac{-11±\sqrt{23}i}{32} when ± is minus. Subtract i\sqrt{23} from -11.

$$x=\frac{-\sqrt{23}i-11}{32}$$

The equation is now solved.

$$x=\frac{-11+\sqrt{23}i}{32} x=\frac{-\sqrt{23}i-11}{32}$$

Substitute \frac{-11+\sqrt{23}i}{32} for x in the equation 4x+\frac{3}{2}=\sqrt{x}.

$$4\times \frac{-11+\sqrt{23}i}{32}+\frac{3}{2}=\sqrt{\frac{-11+\sqrt{23}i}{32}}$$

Simplify. The value x=\frac{-11+\sqrt{23}i}{32} satisfies the equation.

$$\frac{1}{8}+\frac{1}{8}i\times 23^{\frac{1}{2}}=\frac{1}{8}+\frac{1}{8}i\times 23^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{23}i-11}{32} for x in the equation 4x+\frac{3}{2}=\sqrt{x}.

$$4\times \frac{-\sqrt{23}i-11}{32}+\frac{3}{2}=\sqrt{\frac{-\sqrt{23}i-11}{32}}$$

Simplify. The value x=\frac{-\sqrt{23}i-11}{32} does not satisfy the equation.

$$-\frac{1}{8}i\times 23^{\frac{1}{2}}+\frac{1}{8}=-\left(\frac{1}{8}-\frac{1}{8}i\times 23^{\frac{1}{2}}\right)$$

Equation 4x+\frac{3}{2}=\sqrt{x} has a unique solution.

$$x=\frac{-11+\sqrt{23}i}{32}$$