Solve sqrt{0.5^2-left(x-0.5right)^2}+0.5=1.25-x

$$\sqrt{ { 0.5 }^{ 2 } - { \left(x-0.5 \right) }^{ 2 } } +0.5 = 1.25-x$$

Solve for x

$$x=\frac{5-\sqrt{7}}{8}\approx 0.294281086$$

Solution Steps

Subtract 0.5 from both sides of the equation.

$$\sqrt{0.5^{2}-\left(x-0.5\right)^{2}}=1.25-x-0.5$$

Calculate 0.5 to the power of 2 and get 0.25.

$$\sqrt{0.25-\left(x-0.5\right)^{2}}=1.25-x-0.5$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-0.5\right)^{2}.

$$\sqrt{0.25-\left(x^{2}-x+0.25\right)}=1.25-x-0.5$$

To find the opposite of x^{2}-x+0.25, find the opposite of each term.

$$\sqrt{0.25-x^{2}+x-0.25}=1.25-x-0.5$$

Subtract 0.25 from 0.25 to get 0.

$$\sqrt{-x^{2}+x}=1.25-x-0.5$$

Subtract 0.5 from 1.25 to get 0.75.

$$\sqrt{-x^{2}+x}=0.75-x$$

Square both sides of the equation.

$$\left(\sqrt{-x^{2}+x}\right)^{2}=\left(0.75-x\right)^{2}$$

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

$$-x^{2}+x=\left(0.75-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(0.75-x\right)^{2}.

$$-x^{2}+x=0.5625-1.5x+x^{2}$$

Subtract 0.5625 from both sides.

$$-x^{2}+x-0.5625=-1.5x+x^{2}$$

Add 1.5x to both sides.

$$-x^{2}+x-0.5625+1.5x=x^{2}$$

Combine x and 1.5x to get 2.5x.

$$-x^{2}+2.5x-0.5625=x^{2}$$

Subtract x^{2} from both sides.

$$-x^{2}+2.5x-0.5625-x^{2}=0$$

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

$$-2x^{2}+2.5x-0.5625=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 2.5 for b, and -0.5625 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2.5±\sqrt{2.5^{2}-4\left(-2\right)\left(-0.5625\right)}}{2\left(-2\right)}$$

Square 2.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-2.5±\sqrt{6.25-4\left(-2\right)\left(-0.5625\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-2.5±\sqrt{6.25+8\left(-0.5625\right)}}{2\left(-2\right)}$$

Multiply 8 times -0.5625.

$$x=\frac{-2.5±\sqrt{6.25-4.5}}{2\left(-2\right)}$$

Add 6.25 to -4.5 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-2.5±\sqrt{1.75}}{2\left(-2\right)}$$

Take the square root of 1.75.

$$x=\frac{-2.5±\frac{\sqrt{7}}{2}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-2.5±\frac{\sqrt{7}}{2}}{-4}$$

Now solve the equation x=\frac{-2.5±\frac{\sqrt{7}}{2}}{-4} when ± is plus. Add -2.5 to \frac{\sqrt{7}}{2}.

$$x=\frac{\sqrt{7}-5}{-4\times 2}$$

Divide \frac{-5+\sqrt{7}}{2} by -4.

$$x=\frac{5-\sqrt{7}}{8}$$

Now solve the equation x=\frac{-2.5±\frac{\sqrt{7}}{2}}{-4} when ± is minus. Subtract \frac{\sqrt{7}}{2} from -2.5.

$$x=\frac{-\sqrt{7}-5}{-4\times 2}$$

Divide \frac{-5-\sqrt{7}}{2} by -4.

$$x=\frac{\sqrt{7}+5}{8}$$

The equation is now solved.

$$x=\frac{5-\sqrt{7}}{8} x=\frac{\sqrt{7}+5}{8}$$

Substitute \frac{5-\sqrt{7}}{8} for x in the equation \sqrt{0.5^{2}-\left(x-0.5\right)^{2}}+0.5=1.25-x.

$$\sqrt{0.5^{2}-\left(\frac{5-\sqrt{7}}{8}-0.5\right)^{2}}+0.5=1.25-\frac{5-\sqrt{7}}{8}$$

Simplify. The value x=\frac{5-\sqrt{7}}{8} satisfies the equation.

$$\frac{1}{8}\times 7^{\frac{1}{2}}+\frac{5}{8}=\frac{5}{8}+\frac{1}{8}\times 7^{\frac{1}{2}}$$

Substitute \frac{\sqrt{7}+5}{8} for x in the equation \sqrt{0.5^{2}-\left(x-0.5\right)^{2}}+0.5=1.25-x.

$$\sqrt{0.5^{2}-\left(\frac{\sqrt{7}+5}{8}-0.5\right)^{2}}+0.5=1.25-\frac{\sqrt{7}+5}{8}$$

Simplify. The value x=\frac{\sqrt{7}+5}{8} does not satisfy the equation.

$$\frac{1}{8}\times 7^{\frac{1}{2}}+\frac{3}{8}=\frac{5}{8}-\frac{1}{8}\times 7^{\frac{1}{2}}$$

Equation \sqrt{x-x^{2}}=0.75-x has a unique solution.

$$x=\frac{5-\sqrt{7}}{8}$$