Solve x-7=sqrt{3x-72}

$$x - 7 = \sqrt { 3 x - 72 }$$

Solve for x (complex solution)

$$x=\frac{17+\sqrt{195}i}{2}\approx 8.5+6.982120022i$$

Solution Steps

Square both sides of the equation.

$$\left(x-7\right)^{2}=\left(\sqrt{3x-72}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-7\right)^{2}.

$$x^{2}-14x+49=\left(\sqrt{3x-72}\right)^{2}$$

Calculate \sqrt{3x-72} to the power of 2 and get 3x-72.

$$x^{2}-14x+49=3x-72$$

Subtract 3x from both sides.

$$x^{2}-14x+49-3x=-72$$

Combine -14x and -3x to get -17x.

$$x^{2}-17x+49=-72$$

Add 72 to both sides.

$$x^{2}-17x+49+72=0$$

Add 49 and 72 to get 121.

$$x^{2}-17x+121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -17 for b, and 121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 121}}{2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 121}}{2}$$

Multiply -4 times 121.

$$x=\frac{-\left(-17\right)±\sqrt{289-484}}{2}$$

Add 289 to -484.

$$x=\frac{-\left(-17\right)±\sqrt{-195}}{2}$$

Take the square root of -195.

$$x=\frac{-\left(-17\right)±\sqrt{195}i}{2}$$

The opposite of -17 is 17.

$$x=\frac{17±\sqrt{195}i}{2}$$

Now solve the equation x=\frac{17±\sqrt{195}i}{2} when ± is plus. Add 17 to i\sqrt{195}.

$$x=\frac{17+\sqrt{195}i}{2}$$

Now solve the equation x=\frac{17±\sqrt{195}i}{2} when ± is minus. Subtract i\sqrt{195} from 17.

$$x=\frac{-\sqrt{195}i+17}{2}$$

The equation is now solved.

$$x=\frac{17+\sqrt{195}i}{2} x=\frac{-\sqrt{195}i+17}{2}$$

Substitute \frac{17+\sqrt{195}i}{2} for x in the equation x-7=\sqrt{3x-72}.

$$\frac{17+\sqrt{195}i}{2}-7=\sqrt{3\times \frac{17+\sqrt{195}i}{2}-72}$$

Simplify. The value x=\frac{17+\sqrt{195}i}{2} satisfies the equation.

$$\frac{3}{2}+\frac{1}{2}i\times 195^{\frac{1}{2}}=\frac{3}{2}+\frac{1}{2}i\times 195^{\frac{1}{2}}$$

Substitute \frac{-\sqrt{195}i+17}{2} for x in the equation x-7=\sqrt{3x-72}.

$$\frac{-\sqrt{195}i+17}{2}-7=\sqrt{3\times \frac{-\sqrt{195}i+17}{2}-72}$$

Simplify. The value x=\frac{-\sqrt{195}i+17}{2} does not satisfy the equation.

$$-\frac{1}{2}i\times 195^{\frac{1}{2}}+\frac{3}{2}=-\left(\frac{3}{2}-\frac{1}{2}i\times 195^{\frac{1}{2}}\right)$$

Equation x-7=\sqrt{3x-72} has a unique solution.

$$x=\frac{17+\sqrt{195}i}{2}$$