Solve x-sqrt{9-3x}=0

$$x- \sqrt{ 9-3x } =0$$

Solve for x

$$x = \frac{3 {(\sqrt{5} - 1)}}{2} \approx 1.854101966$$

Solution Steps

Subtract x from both sides of the equation.

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

Cancel out -1 on both sides.

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

Square both sides of the equation.

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

Calculate \sqrt{9-3x} to the power of 2 and get 9-3x.

$$9-3x=x^{2}$$

Subtract x^{2} from both sides.

$$9-3x-x^{2}=0$$

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.

$$-x^{2}-3x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -3 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 9}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 9}}{2\left(-1\right)}$$

Multiply 4 times 9.

$$x=\frac{-\left(-3\right)±\sqrt{9+36}}{2\left(-1\right)}$$

Add 9 to 36.

$$x=\frac{-\left(-3\right)±\sqrt{45}}{2\left(-1\right)}$$

Take the square root of 45.

$$x=\frac{-\left(-3\right)±3\sqrt{5}}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±3\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{3±3\sqrt{5}}{-2}$$

Now solve the equation x=\frac{3±3\sqrt{5}}{-2} when ± is plus. Add 3 to 3\sqrt{5}.

$$x=\frac{3\sqrt{5}+3}{-2}$$

Divide 3+3\sqrt{5} by -2.

$$x=\frac{-3\sqrt{5}-3}{2}$$

Now solve the equation x=\frac{3±3\sqrt{5}}{-2} when ± is minus. Subtract 3\sqrt{5} from 3.

$$x=\frac{3-3\sqrt{5}}{-2}$$

Divide 3-3\sqrt{5} by -2.

$$x=\frac{3\sqrt{5}-3}{2}$$

The equation is now solved.

$$x=\frac{-3\sqrt{5}-3}{2} x=\frac{3\sqrt{5}-3}{2}$$

Substitute \frac{-3\sqrt{5}-3}{2} for x in the equation x-\sqrt{9-3x}=0.

$$\frac{-3\sqrt{5}-3}{2}-\sqrt{9-3\times \frac{-3\sqrt{5}-3}{2}}=0$$

Simplify. The value x=\frac{-3\sqrt{5}-3}{2} does not satisfy the equation.

$$-3\times 5^{\frac{1}{2}}-3=0$$

Substitute \frac{3\sqrt{5}-3}{2} for x in the equation x-\sqrt{9-3x}=0.

$$\frac{3\sqrt{5}-3}{2}-\sqrt{9-3\times \frac{3\sqrt{5}-3}{2}}=0$$

Simplify. The value x=\frac{3\sqrt{5}-3}{2} satisfies the equation.

$$0=0$$

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

$$x=\frac{3\sqrt{5}-3}{2}$$