Solve sqrt{2x-3}=6^2x*sqrt{4}

$$\sqrt { 2 x - 3 } = 6 ^ { 2 } x \cdot \sqrt { 4 }$$

Solve for x (complex solution)

$$x=\frac{1+\sqrt{15551}i}{5184}\approx 0.000192901+0.024055488i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{2x-3}\right)^{2}=\left(6^{2}x\sqrt{4}\right)^{2}$$

Calculate \sqrt{2x-3} to the power of 2 and get 2x-3.

$$2x-3=\left(6^{2}x\sqrt{4}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$2x-3=\left(36x\sqrt{4}\right)^{2}$$

Calculate the square root of 4 and get 2.

$$2x-3=\left(36x\times 2\right)^{2}$$

Multiply 36 and 2 to get 72.

$$2x-3=\left(72x\right)^{2}$$

Expand \left(72x\right)^{2}.

$$2x-3=72^{2}x^{2}$$

Calculate 72 to the power of 2 and get 5184.

$$2x-3=5184x^{2}$$

Subtract 5184x^{2} from both sides.

$$2x-3-5184x^{2}=0$$

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.

$$-5184x^{2}+2x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5184 for a, 2 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-5184\right)\left(-3\right)}}{2\left(-5184\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-5184\right)\left(-3\right)}}{2\left(-5184\right)}$$

Multiply -4 times -5184.

$$x=\frac{-2±\sqrt{4+20736\left(-3\right)}}{2\left(-5184\right)}$$

Multiply 20736 times -3.

$$x=\frac{-2±\sqrt{4-62208}}{2\left(-5184\right)}$$

Add 4 to -62208.

$$x=\frac{-2±\sqrt{-62204}}{2\left(-5184\right)}$$

Take the square root of -62204.

$$x=\frac{-2±2\sqrt{15551}i}{2\left(-5184\right)}$$

Multiply 2 times -5184.

$$x=\frac{-2±2\sqrt{15551}i}{-10368}$$

Now solve the equation x=\frac{-2±2\sqrt{15551}i}{-10368} when ± is plus. Add -2 to 2i\sqrt{15551}.

$$x=\frac{-2+2\sqrt{15551}i}{-10368}$$

Divide -2+2i\sqrt{15551} by -10368.

$$x=\frac{-\sqrt{15551}i+1}{5184}$$

Now solve the equation x=\frac{-2±2\sqrt{15551}i}{-10368} when ± is minus. Subtract 2i\sqrt{15551} from -2.

$$x=\frac{-2\sqrt{15551}i-2}{-10368}$$

Divide -2-2i\sqrt{15551} by -10368.

$$x=\frac{1+\sqrt{15551}i}{5184}$$

The equation is now solved.

$$x=\frac{-\sqrt{15551}i+1}{5184} x=\frac{1+\sqrt{15551}i}{5184}$$

Substitute \frac{-\sqrt{15551}i+1}{5184} for x in the equation \sqrt{2x-3}=6^{2}x\sqrt{4}.

$$\sqrt{2\times \frac{-\sqrt{15551}i+1}{5184}-3}=6^{2}\times \frac{-\sqrt{15551}i+1}{5184}\sqrt{4}$$

Simplify. The value x=\frac{-\sqrt{15551}i+1}{5184} does not satisfy the equation.

$$-\left(\frac{1}{72}-\frac{1}{72}i\times 15551^{\frac{1}{2}}\right)=-\frac{1}{72}i\times 15551^{\frac{1}{2}}+\frac{1}{72}$$

Substitute \frac{1+\sqrt{15551}i}{5184} for x in the equation \sqrt{2x-3}=6^{2}x\sqrt{4}.

$$\sqrt{2\times \frac{1+\sqrt{15551}i}{5184}-3}=6^{2}\times \frac{1+\sqrt{15551}i}{5184}\sqrt{4}$$

Simplify. The value x=\frac{1+\sqrt{15551}i}{5184} satisfies the equation.

$$\frac{1}{72}+\frac{1}{72}i\times 15551^{\frac{1}{2}}=\frac{1}{72}+\frac{1}{72}i\times 15551^{\frac{1}{2}}$$

Equation \sqrt{2x-3}=36\sqrt{4}x has a unique solution.

$$x=\frac{1+\sqrt{15551}i}{5184}$$