Solve sqrt{x}=x-4

$$\sqrt{ x } =x-4$$

Solve for x

$$x = \frac{\sqrt{17} + 9}{2} \approx 6.561552813$$

Solution Steps

Square both sides of the equation.

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

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

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

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

$$x=x^{2}-8x+16$$

Subtract x^{2} from both sides.

$$x-x^{2}=-8x+16$$

Add 8x to both sides.

$$x-x^{2}+8x=16$$

Combine x and 8x to get 9x.

$$9x-x^{2}=16$$

Subtract 16 from both sides.

$$9x-x^{2}-16=0$$

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.

$$-x^{2}+9x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 9 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-16\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-9±\sqrt{81+4\left(-16\right)}}{2\left(-1\right)}$$

Multiply 4 times -16.

$$x=\frac{-9±\sqrt{81-64}}{2\left(-1\right)}$$

Add 81 to -64.

$$x=\frac{-9±\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-9±\sqrt{17}}{-2}$$

Now solve the equation x=\frac{-9±\sqrt{17}}{-2} when ± is plus. Add -9 to \sqrt{17}.

$$x=\frac{\sqrt{17}-9}{-2}$$

Divide -9+\sqrt{17} by -2.

$$x=\frac{9-\sqrt{17}}{2}$$

Now solve the equation x=\frac{-9±\sqrt{17}}{-2} when ± is minus. Subtract \sqrt{17} from -9.

$$x=\frac{-\sqrt{17}-9}{-2}$$

Divide -9-\sqrt{17} by -2.

$$x=\frac{\sqrt{17}+9}{2}$$

The equation is now solved.

$$x=\frac{9-\sqrt{17}}{2} x=\frac{\sqrt{17}+9}{2}$$

Substitute \frac{9-\sqrt{17}}{2} for x in the equation \sqrt{x}=x-4.

$$\sqrt{\frac{9-\sqrt{17}}{2}}=\frac{9-\sqrt{17}}{2}-4$$

Simplify. The value x=\frac{9-\sqrt{17}}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 17^{\frac{1}{2}}\right)=\frac{1}{2}-\frac{1}{2}\times 17^{\frac{1}{2}}$$

Substitute \frac{\sqrt{17}+9}{2} for x in the equation \sqrt{x}=x-4.

$$\sqrt{\frac{\sqrt{17}+9}{2}}=\frac{\sqrt{17}+9}{2}-4$$

Simplify. The value x=\frac{\sqrt{17}+9}{2} satisfies the equation.

$$\frac{1}{2}+\frac{1}{2}\times 17^{\frac{1}{2}}=\frac{1}{2}\times 17^{\frac{1}{2}}+\frac{1}{2}$$

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

$$x=\frac{\sqrt{17}+9}{2}$$