Solve sqrt{x}=180^circ-x

$$\sqrt { x } = 180 ^ { \circ } - x$$

Solve for x

$$x = \frac{361 - \sqrt{721}}{2} \approx 167.074278418$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(180-x\right)^{2}$$

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

$$x=\left(180-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(180-x\right)^{2}.

$$x=32400-360x+x^{2}$$

Subtract 32400 from both sides.

$$x-32400=-360x+x^{2}$$

Add 360x to both sides.

$$x-32400+360x=x^{2}$$

Combine x and 360x to get 361x.

$$361x-32400=x^{2}$$

Subtract x^{2} from both sides.

$$361x-32400-x^{2}=0$$

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.

$$-x^{2}+361x-32400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 361 for b, and -32400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-361±\sqrt{361^{2}-4\left(-1\right)\left(-32400\right)}}{2\left(-1\right)}$$

Square 361.

$$x=\frac{-361±\sqrt{130321-4\left(-1\right)\left(-32400\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-361±\sqrt{130321+4\left(-32400\right)}}{2\left(-1\right)}$$

Multiply 4 times -32400.

$$x=\frac{-361±\sqrt{130321-129600}}{2\left(-1\right)}$$

Add 130321 to -129600.

$$x=\frac{-361±\sqrt{721}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-361±\sqrt{721}}{-2}$$

Now solve the equation x=\frac{-361±\sqrt{721}}{-2} when ± is plus. Add -361 to \sqrt{721}.

$$x=\frac{\sqrt{721}-361}{-2}$$

Divide -361+\sqrt{721} by -2.

$$x=\frac{361-\sqrt{721}}{2}$$

Now solve the equation x=\frac{-361±\sqrt{721}}{-2} when ± is minus. Subtract \sqrt{721} from -361.

$$x=\frac{-\sqrt{721}-361}{-2}$$

Divide -361-\sqrt{721} by -2.

$$x=\frac{\sqrt{721}+361}{2}$$

The equation is now solved.

$$x=\frac{361-\sqrt{721}}{2} x=\frac{\sqrt{721}+361}{2}$$

Substitute \frac{361-\sqrt{721}}{2} for x in the equation \sqrt{x}=180-x.

$$\sqrt{\frac{361-\sqrt{721}}{2}}=180-\frac{361-\sqrt{721}}{2}$$

Simplify. The value x=\frac{361-\sqrt{721}}{2} satisfies the equation.

$$-\left(\frac{1}{2}-\frac{1}{2}\times 721^{\frac{1}{2}}\right)=-\frac{1}{2}+\frac{1}{2}\times 721^{\frac{1}{2}}$$

Substitute \frac{\sqrt{721}+361}{2} for x in the equation \sqrt{x}=180-x.

$$\sqrt{\frac{\sqrt{721}+361}{2}}=180-\frac{\sqrt{721}+361}{2}$$

Simplify. The value x=\frac{\sqrt{721}+361}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{2}+\frac{1}{2}\times 721^{\frac{1}{2}}=-\frac{1}{2}-\frac{1}{2}\times 721^{\frac{1}{2}}$$

Equation \sqrt{x}=180-x has a unique solution.

$$x=\frac{361-\sqrt{721}}{2}$$