Solve 3/8+1/2x=2x

$$\frac{ 3 }{ 8 } + \frac{ 1 }{ 2x } =2x$$

Solve for x

$$x=\frac{3-\sqrt{265}}{32}\approx -0.414963144$$

$$x=\frac{\sqrt{265}+3}{32}\approx 0.602463144$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8x, the least common multiple of 8,2x.

$$8x\times \frac{3}{8}+4=2x\times 8x$$

Multiply 8 and \frac{3}{8} to get 3.

$$3x+4=2x\times 8x$$

Multiply x and x to get x^{2}.

$$3x+4=2x^{2}\times 8$$

Multiply 2 and 8 to get 16.

$$3x+4=16x^{2}$$

Subtract 16x^{2} from both sides.

$$3x+4-16x^{2}=0$$

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.

$$-16x^{2}+3x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 3 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-16\right)\times 4}}{2\left(-16\right)}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\left(-16\right)\times 4}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$x=\frac{-3±\sqrt{9+64\times 4}}{2\left(-16\right)}$$

Multiply 64 times 4.

$$x=\frac{-3±\sqrt{9+256}}{2\left(-16\right)}$$

Add 9 to 256.

$$x=\frac{-3±\sqrt{265}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$x=\frac{-3±\sqrt{265}}{-32}$$

Now solve the equation x=\frac{-3±\sqrt{265}}{-32} when ± is plus. Add -3 to \sqrt{265}.

$$x=\frac{\sqrt{265}-3}{-32}$$

Divide -3+\sqrt{265} by -32.

$$x=\frac{3-\sqrt{265}}{32}$$

Now solve the equation x=\frac{-3±\sqrt{265}}{-32} when ± is minus. Subtract \sqrt{265} from -3.

$$x=\frac{-\sqrt{265}-3}{-32}$$

Divide -3-\sqrt{265} by -32.

$$x=\frac{\sqrt{265}+3}{32}$$

The equation is now solved.

$$x=\frac{3-\sqrt{265}}{32} x=\frac{\sqrt{265}+3}{32}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 8x, the least common multiple of 8,2x.

$$8x\times \frac{3}{8}+4=2x\times 8x$$

Multiply 8 and \frac{3}{8} to get 3.

$$3x+4=2x\times 8x$$

Multiply x and x to get x^{2}.

$$3x+4=2x^{2}\times 8$$

Multiply 2 and 8 to get 16.

$$3x+4=16x^{2}$$

Subtract 16x^{2} from both sides.

$$3x+4-16x^{2}=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$3x-16x^{2}=-4$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-16x^{2}+3x=-4$$

Divide both sides by -16.

$$\frac{-16x^{2}+3x}{-16}=-\frac{4}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$x^{2}+\frac{3}{-16}x=-\frac{4}{-16}$$

Divide 3 by -16.

$$x^{2}-\frac{3}{16}x=-\frac{4}{-16}$$

Reduce the fraction \frac{-4}{-16} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3}{16}x=\frac{1}{4}$$

Divide -\frac{3}{16}, the coefficient of the x term, by 2 to get -\frac{3}{32}. Then add the square of -\frac{3}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3}{16}x+\left(-\frac{3}{32}\right)^{2}=\frac{1}{4}+\left(-\frac{3}{32}\right)^{2}$$

Square -\frac{3}{32} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3}{16}x+\frac{9}{1024}=\frac{1}{4}+\frac{9}{1024}$$

Add \frac{1}{4} to \frac{9}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3}{16}x+\frac{9}{1024}=\frac{265}{1024}$$

Factor x^{2}-\frac{3}{16}x+\frac{9}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3}{32}\right)^{2}=\frac{265}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3}{32}\right)^{2}}=\sqrt{\frac{265}{1024}}$$

Simplify.

$$x-\frac{3}{32}=\frac{\sqrt{265}}{32} x-\frac{3}{32}=-\frac{\sqrt{265}}{32}$$

Add \frac{3}{32} to both sides of the equation.

$$x=\frac{\sqrt{265}+3}{32} x=\frac{3-\sqrt{265}}{32}$$