Solve sqrt{(5/2)^2-(x-3/2)^2}=-2x

$$\sqrt { ( \frac { 5 } { 2 } ) ^ { 2 } - ( x - \frac { 3 } { 2 } ) ^ { 2 } } = - 2 x$$

Solve for x

$$x=\frac{3-\sqrt{89}}{10}\approx -0.643398113$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\left(\frac{5}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}}\right)^{2}=\left(-2x\right)^{2}$$

Calculate \frac{5}{2} to the power of 2 and get \frac{25}{4}.

$$\left(\sqrt{\frac{25}{4}-\left(x-\frac{3}{2}\right)^{2}}\right)^{2}=\left(-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{3}{2}\right)^{2}.

$$\left(\sqrt{\frac{25}{4}-\left(x^{2}-3x+\frac{9}{4}\right)}\right)^{2}=\left(-2x\right)^{2}$$

To find the opposite of x^{2}-3x+\frac{9}{4}, find the opposite of each term.

$$\left(\sqrt{\frac{25}{4}-x^{2}+3x-\frac{9}{4}}\right)^{2}=\left(-2x\right)^{2}$$

Subtract \frac{9}{4} from \frac{25}{4} to get 4.

$$\left(\sqrt{4-x^{2}+3x}\right)^{2}=\left(-2x\right)^{2}$$

Calculate \sqrt{4-x^{2}+3x} to the power of 2 and get 4-x^{2}+3x.

$$4-x^{2}+3x=\left(-2x\right)^{2}$$

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

$$4-x^{2}+3x=\left(-2\right)^{2}x^{2}$$

Calculate -2 to the power of 2 and get 4.

$$4-x^{2}+3x=4x^{2}$$

Subtract 4x^{2} from both sides.

$$4-x^{2}+3x-4x^{2}=0$$

Combine -x^{2} and -4x^{2} to get -5x^{2}.

$$4-5x^{2}+3x=0$$

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.

$$-5x^{2}+3x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 3 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-5\right)\times 4}}{2\left(-5\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-5\right)\times 4}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-3±\sqrt{9+20\times 4}}{2\left(-5\right)}$$

Multiply 20 times 4.

$$x=\frac{-3±\sqrt{9+80}}{2\left(-5\right)}$$

Add 9 to 80.

$$x=\frac{-3±\sqrt{89}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-3±\sqrt{89}}{-10}$$

Now solve the equation x=\frac{-3±\sqrt{89}}{-10} when ± is plus. Add -3 to \sqrt{89}.

$$x=\frac{\sqrt{89}-3}{-10}$$

Divide -3+\sqrt{89} by -10.

$$x=\frac{3-\sqrt{89}}{10}$$

Now solve the equation x=\frac{-3±\sqrt{89}}{-10} when ± is minus. Subtract \sqrt{89} from -3.

$$x=\frac{-\sqrt{89}-3}{-10}$$

Divide -3-\sqrt{89} by -10.

$$x=\frac{\sqrt{89}+3}{10}$$

The equation is now solved.

$$x=\frac{3-\sqrt{89}}{10} x=\frac{\sqrt{89}+3}{10}$$

Substitute \frac{3-\sqrt{89}}{10} for x in the equation \sqrt{\left(\frac{5}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}}=-2x.

$$\sqrt{\left(\frac{5}{2}\right)^{2}-\left(\frac{3-\sqrt{89}}{10}-\frac{3}{2}\right)^{2}}=-2\times \frac{3-\sqrt{89}}{10}$$

Simplify. The value x=\frac{3-\sqrt{89}}{10} satisfies the equation.

$$\frac{1}{5}\times 89^{\frac{1}{2}}-\frac{3}{5}=-\frac{3}{5}+\frac{1}{5}\times 89^{\frac{1}{2}}$$

Substitute \frac{\sqrt{89}+3}{10} for x in the equation \sqrt{\left(\frac{5}{2}\right)^{2}-\left(x-\frac{3}{2}\right)^{2}}=-2x.

$$\sqrt{\left(\frac{5}{2}\right)^{2}-\left(\frac{\sqrt{89}+3}{10}-\frac{3}{2}\right)^{2}}=-2\times \frac{\sqrt{89}+3}{10}$$

Simplify. The value x=\frac{\sqrt{89}+3}{10} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{5}\times 89^{\frac{1}{2}}+\frac{3}{5}=-\frac{1}{5}\times 89^{\frac{1}{2}}-\frac{3}{5}$$

Equation \sqrt{\frac{25}{4}-\left(x-\frac{3}{2}\right)^{2}}=-2x has a unique solution.

$$x=\frac{3-\sqrt{89}}{10}$$