Solve 5x/34.714+.5x-15/x=1

$$\frac { 5 x } { 34.714 } + \frac { .5 x - 15 } { x } = 1$$

Solve for x

$$x = \frac{13 \sqrt{63405121} + 17357}{10000} \approx 12.08725324$$

$$x=\frac{17357-13\sqrt{63405121}}{10000}\approx -8.61585324$$

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 x.

$$x\times \frac{5x}{34.714}+0.5x-15=x$$

Divide 5x by 34.714 to get \frac{2500}{17357}x.

$$x\times \frac{2500}{17357}x+0.5x-15=x$$

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

$$x^{2}\times \frac{2500}{17357}+0.5x-15=x$$

Subtract x from both sides.

$$x^{2}\times \frac{2500}{17357}+0.5x-15-x=0$$

Combine 0.5x and -x to get -0.5x.

$$x^{2}\times \frac{2500}{17357}-0.5x-15=0$$

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.

$$\frac{2500}{17357}x^{2}-0.5x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{2500}{17357} for a, -0.5 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-0.5\right)±\sqrt{\left(-0.5\right)^{2}-4\times \frac{2500}{17357}\left(-15\right)}}{2\times \frac{2500}{17357}}$$

Square -0.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-0.5\right)±\sqrt{0.25-4\times \frac{2500}{17357}\left(-15\right)}}{2\times \frac{2500}{17357}}$$

Multiply -4 times \frac{2500}{17357}.

$$x=\frac{-\left(-0.5\right)±\sqrt{0.25-\frac{10000}{17357}\left(-15\right)}}{2\times \frac{2500}{17357}}$$

Multiply -\frac{10000}{17357} times -15.

$$x=\frac{-\left(-0.5\right)±\sqrt{0.25+\frac{150000}{17357}}}{2\times \frac{2500}{17357}}$$

Add 0.25 to \frac{150000}{17357} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-0.5\right)±\sqrt{\frac{617357}{69428}}}{2\times \frac{2500}{17357}}$$

Take the square root of \frac{617357}{69428}.

$$x=\frac{-\left(-0.5\right)±\frac{13\sqrt{63405121}}{34714}}{2\times \frac{2500}{17357}}$$

The opposite of -0.5 is 0.5.

$$x=\frac{0.5±\frac{13\sqrt{63405121}}{34714}}{2\times \frac{2500}{17357}}$$

Multiply 2 times \frac{2500}{17357}.

$$x=\frac{0.5±\frac{13\sqrt{63405121}}{34714}}{\frac{5000}{17357}}$$

Now solve the equation x=\frac{0.5±\frac{13\sqrt{63405121}}{34714}}{\frac{5000}{17357}} when ± is plus. Add 0.5 to \frac{13\sqrt{63405121}}{34714}.

$$x=\frac{\frac{13\sqrt{63405121}}{34714}+\frac{1}{2}}{\frac{5000}{17357}}$$

Divide \frac{1}{2}+\frac{13\sqrt{63405121}}{34714} by \frac{5000}{17357} by multiplying \frac{1}{2}+\frac{13\sqrt{63405121}}{34714} by the reciprocal of \frac{5000}{17357}.

$$x=\frac{13\sqrt{63405121}+17357}{10000}$$

Now solve the equation x=\frac{0.5±\frac{13\sqrt{63405121}}{34714}}{\frac{5000}{17357}} when ± is minus. Subtract \frac{13\sqrt{63405121}}{34714} from 0.5.

$$x=\frac{-\frac{13\sqrt{63405121}}{34714}+\frac{1}{2}}{\frac{5000}{17357}}$$

Divide \frac{1}{2}-\frac{13\sqrt{63405121}}{34714} by \frac{5000}{17357} by multiplying \frac{1}{2}-\frac{13\sqrt{63405121}}{34714} by the reciprocal of \frac{5000}{17357}.

$$x=\frac{17357-13\sqrt{63405121}}{10000}$$

The equation is now solved.

$$x=\frac{13\sqrt{63405121}+17357}{10000} x=\frac{17357-13\sqrt{63405121}}{10000}$$

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 x.

$$x\times \frac{5x}{34.714}+0.5x-15=x$$

Divide 5x by 34.714 to get \frac{2500}{17357}x.

$$x\times \frac{2500}{17357}x+0.5x-15=x$$

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

$$x^{2}\times \frac{2500}{17357}+0.5x-15=x$$

Subtract x from both sides.

$$x^{2}\times \frac{2500}{17357}+0.5x-15-x=0$$

Combine 0.5x and -x to get -0.5x.

$$x^{2}\times \frac{2500}{17357}-0.5x-15=0$$

Add 15 to both sides. Anything plus zero gives itself.

$$x^{2}\times \frac{2500}{17357}-0.5x=15$$

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.

$$\frac{2500}{17357}x^{2}-0.5x=15$$

Divide both sides of the equation by \frac{2500}{17357}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{2500}{17357}x^{2}-0.5x}{\frac{2500}{17357}}=\frac{15}{\frac{2500}{17357}}$$

Dividing by \frac{2500}{17357} undoes the multiplication by \frac{2500}{17357}.

$$x^{2}+\left(-\frac{0.5}{\frac{2500}{17357}}\right)x=\frac{15}{\frac{2500}{17357}}$$

Divide -0.5 by \frac{2500}{17357} by multiplying -0.5 by the reciprocal of \frac{2500}{17357}.

$$x^{2}-3.4714x=\frac{15}{\frac{2500}{17357}}$$

Divide 15 by \frac{2500}{17357} by multiplying 15 by the reciprocal of \frac{2500}{17357}.

$$x^{2}-3.4714x=104.142$$

Divide -3.4714, the coefficient of the x term, by 2 to get -1.7357. Then add the square of -1.7357 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-3.4714x+\left(-1.7357\right)^{2}=104.142+\left(-1.7357\right)^{2}$$

Square -1.7357 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-3.4714x+3.01265449=104.142+3.01265449$$

Add 104.142 to 3.01265449 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-3.4714x+3.01265449=107.15465449$$

Factor x^{2}-3.4714x+3.01265449. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1.7357\right)^{2}=107.15465449$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1.7357\right)^{2}}=\sqrt{107.15465449}$$

Simplify.

$$x-1.7357=\frac{13\sqrt{63405121}}{10000} x-1.7357=-\frac{13\sqrt{63405121}}{10000}$$

Add 1.7357 to both sides of the equation.

$$x=\frac{13\sqrt{63405121}+17357}{10000} x=\frac{17357-13\sqrt{63405121}}{10000}$$