Solve x-sqrt{x}-72=0

$$x- \sqrt{ x } -72 = 0$$

Solve for x

$$x=81$$

Solution Steps

Subtract x-72 from both sides of the equation.

$$-\sqrt{x}=-\left(x-72\right)$$

Cancel out -1 on both sides.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-72\right)^{2}.

$$x=x^{2}-144x+5184$$

Subtract x^{2} from both sides.

$$x-x^{2}=-144x+5184$$

Add 144x to both sides.

$$x-x^{2}+144x=5184$$

Combine x and 144x to get 145x.

$$145x-x^{2}=5184$$

Subtract 5184 from both sides.

$$145x-x^{2}-5184=0$$

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.

$$-x^{2}+145x-5184=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 145 for b, and -5184 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-145±\sqrt{145^{2}-4\left(-1\right)\left(-5184\right)}}{2\left(-1\right)}$$

Square 145.

$$x=\frac{-145±\sqrt{21025-4\left(-1\right)\left(-5184\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-145±\sqrt{21025+4\left(-5184\right)}}{2\left(-1\right)}$$

Multiply 4 times -5184.

$$x=\frac{-145±\sqrt{21025-20736}}{2\left(-1\right)}$$

Add 21025 to -20736.

$$x=\frac{-145±\sqrt{289}}{2\left(-1\right)}$$

Take the square root of 289.

$$x=\frac{-145±17}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-145±17}{-2}$$

Now solve the equation x=\frac{-145±17}{-2} when ± is plus. Add -145 to 17.

$$x=-\frac{128}{-2}$$

Divide -128 by -2.

$$x=64$$

Now solve the equation x=\frac{-145±17}{-2} when ± is minus. Subtract 17 from -145.

$$x=-\frac{162}{-2}$$

Divide -162 by -2.

$$x=81$$

The equation is now solved.

$$x=64 x=81$$

Substitute 64 for x in the equation x-\sqrt{x}-72=0.

$$64-\sqrt{64}-72=0$$

Simplify. The value x=64 does not satisfy the equation.

$$-16=0$$

Substitute 81 for x in the equation x-\sqrt{x}-72=0.

$$81-\sqrt{81}-72=0$$

Simplify. The value x=81 satisfies the equation.

$$0=0$$

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

$$x=81$$