Solve x-sqrt{25-x^2}=2

$$x - \sqrt { 25 - x ^ { 2 } } = 2$$

Solve for x

$$x = \frac{\sqrt{46} + 2}{2} \approx 4.391164992$$

Solution Steps

Subtract x from both sides of the equation.

$$-\sqrt{25-x^{2}}=2-x$$

Square both sides of the equation.

$$\left(-\sqrt{25-x^{2}}\right)^{2}=\left(2-x\right)^{2}$$

Expand \left(-\sqrt{25-x^{2}}\right)^{2}.

$$\left(-1\right)^{2}\left(\sqrt{25-x^{2}}\right)^{2}=\left(2-x\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1\left(\sqrt{25-x^{2}}\right)^{2}=\left(2-x\right)^{2}$$

Calculate \sqrt{25-x^{2}} to the power of 2 and get 25-x^{2}.

$$1\left(25-x^{2}\right)=\left(2-x\right)^{2}$$

Use the distributive property to multiply 1 by 25-x^{2}.

$$25-x^{2}=\left(2-x\right)^{2}$$

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

$$25-x^{2}=4-4x+x^{2}$$

Subtract 4 from both sides.

$$25-x^{2}-4=-4x+x^{2}$$

Subtract 4 from 25 to get 21.

$$21-x^{2}=-4x+x^{2}$$

Add 4x to both sides.

$$21-x^{2}+4x=x^{2}$$

Subtract x^{2} from both sides.

$$21-x^{2}+4x-x^{2}=0$$

Combine -x^{2} and -x^{2} to get -2x^{2}.

$$21-2x^{2}+4x=0$$

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.

$$-2x^{2}+4x+21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 4 for b, and 21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-2\right)\times 21}}{2\left(-2\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 21}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-4±\sqrt{16+8\times 21}}{2\left(-2\right)}$$

Multiply 8 times 21.

$$x=\frac{-4±\sqrt{16+168}}{2\left(-2\right)}$$

Add 16 to 168.

$$x=\frac{-4±\sqrt{184}}{2\left(-2\right)}$$

Take the square root of 184.

$$x=\frac{-4±2\sqrt{46}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-4±2\sqrt{46}}{-4}$$

Now solve the equation x=\frac{-4±2\sqrt{46}}{-4} when ± is plus. Add -4 to 2\sqrt{46}.

$$x=\frac{2\sqrt{46}-4}{-4}$$

Divide -4+2\sqrt{46} by -4.

$$x=-\frac{\sqrt{46}}{2}+1$$

Now solve the equation x=\frac{-4±2\sqrt{46}}{-4} when ± is minus. Subtract 2\sqrt{46} from -4.

$$x=\frac{-2\sqrt{46}-4}{-4}$$

Divide -4-2\sqrt{46} by -4.

$$x=\frac{\sqrt{46}}{2}+1$$

The equation is now solved.

$$x=-\frac{\sqrt{46}}{2}+1 x=\frac{\sqrt{46}}{2}+1$$

Substitute -\frac{\sqrt{46}}{2}+1 for x in the equation x-\sqrt{25-x^{2}}=2.

$$-\frac{\sqrt{46}}{2}+1-\sqrt{25-\left(-\frac{\sqrt{46}}{2}+1\right)^{2}}=2$$

Simplify. The value x=-\frac{\sqrt{46}}{2}+1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-46^{\frac{1}{2}}=2$$

Substitute \frac{\sqrt{46}}{2}+1 for x in the equation x-\sqrt{25-x^{2}}=2.

$$\frac{\sqrt{46}}{2}+1-\sqrt{25-\left(\frac{\sqrt{46}}{2}+1\right)^{2}}=2$$

Simplify. The value x=\frac{\sqrt{46}}{2}+1 satisfies the equation.

$$2=2$$

Equation -\sqrt{25-x^{2}}=2-x has a unique solution.

$$x=\frac{\sqrt{46}}{2}+1$$