Solve sqrt{x-9}=x-3

$$\sqrt { x - 9 } = x - 3$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{23}i}{2}\approx 3.5+2.397915762i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x-9}\right)^{2}=\left(x-3\right)^{2}$$

Calculate \sqrt{x-9} to the power of 2 and get x-9.

$$x-9=\left(x-3\right)^{2}$$

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

$$x-9=x^{2}-6x+9$$

Subtract x^{2} from both sides.

$$x-9-x^{2}=-6x+9$$

Add 6x to both sides.

$$x-9-x^{2}+6x=9$$

Combine x and 6x to get 7x.

$$7x-9-x^{2}=9$$

Subtract 9 from both sides.

$$7x-9-x^{2}-9=0$$

Subtract 9 from -9 to get -18.

$$7x-18-x^{2}=0$$

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.

$$-x^{2}+7x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 7 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-1\right)\left(-18\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-7±\sqrt{49+4\left(-18\right)}}{2\left(-1\right)}$$

Multiply 4 times -18.

$$x=\frac{-7±\sqrt{49-72}}{2\left(-1\right)}$$

Add 49 to -72.

$$x=\frac{-7±\sqrt{-23}}{2\left(-1\right)}$$

Take the square root of -23.

$$x=\frac{-7±\sqrt{23}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-7±\sqrt{23}i}{-2}$$

Now solve the equation x=\frac{-7±\sqrt{23}i}{-2} when ± is plus. Add -7 to i\sqrt{23}.

$$x=\frac{-7+\sqrt{23}i}{-2}$$

Divide -7+i\sqrt{23} by -2.

$$x=\frac{-\sqrt{23}i+7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{23}i}{-2} when ± is minus. Subtract i\sqrt{23} from -7.

$$x=\frac{-\sqrt{23}i-7}{-2}$$

Divide -7-i\sqrt{23} by -2.

$$x=\frac{7+\sqrt{23}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{23}i+7}{2} x=\frac{7+\sqrt{23}i}{2}$$

Substitute \frac{-\sqrt{23}i+7}{2} for x in the equation \sqrt{x-9}=x-3.

$$\sqrt{\frac{-\sqrt{23}i+7}{2}-9}=\frac{-\sqrt{23}i+7}{2}-3$$

Simplify. The value x=\frac{-\sqrt{23}i+7}{2} does not satisfy the equation.

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

Substitute \frac{7+\sqrt{23}i}{2} for x in the equation \sqrt{x-9}=x-3.

$$\sqrt{\frac{7+\sqrt{23}i}{2}-9}=\frac{7+\sqrt{23}i}{2}-3$$

Simplify. The value x=\frac{7+\sqrt{23}i}{2} satisfies the equation.

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

Equation \sqrt{x-9}=x-3 has a unique solution.

$$x=\frac{7+\sqrt{23}i}{2}$$