Solve x/-5+3/x=41/4

$$\frac { x } { - 5 } + \frac { 3 } { x } = 4 \frac { 1 } { 4 }$$

Solve for x

$$x=\frac{\sqrt{8185}-85}{8}\approx 0.683873728$$

$$x=\frac{-\sqrt{8185}-85}{8}\approx -21.933873728$$

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 20x, the least common multiple of -5,x,4.

$$-4xx+20\times 3=5x\left(4\times 4+1\right)$$

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

$$-4x^{2}+20\times 3=5x\left(4\times 4+1\right)$$

Multiply 20 and 3 to get 60.

$$-4x^{2}+60=5x\left(4\times 4+1\right)$$

Multiply 4 and 4 to get 16.

$$-4x^{2}+60=5x\left(16+1\right)$$

Add 16 and 1 to get 17.

$$-4x^{2}+60=5x\times 17$$

Multiply 5 and 17 to get 85.

$$-4x^{2}+60=85x$$

Subtract 85x from both sides.

$$-4x^{2}+60-85x=0$$

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.

$$-4x^{2}-85x+60=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -85 for b, and 60 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-85\right)±\sqrt{\left(-85\right)^{2}-4\left(-4\right)\times 60}}{2\left(-4\right)}$$

Square -85.

$$x=\frac{-\left(-85\right)±\sqrt{7225-4\left(-4\right)\times 60}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\left(-85\right)±\sqrt{7225+16\times 60}}{2\left(-4\right)}$$

Multiply 16 times 60.

$$x=\frac{-\left(-85\right)±\sqrt{7225+960}}{2\left(-4\right)}$$

Add 7225 to 960.

$$x=\frac{-\left(-85\right)±\sqrt{8185}}{2\left(-4\right)}$$

The opposite of -85 is 85.

$$x=\frac{85±\sqrt{8185}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{85±\sqrt{8185}}{-8}$$

Now solve the equation x=\frac{85±\sqrt{8185}}{-8} when ± is plus. Add 85 to \sqrt{8185}.

$$x=\frac{\sqrt{8185}+85}{-8}$$

Divide 85+\sqrt{8185} by -8.

$$x=\frac{-\sqrt{8185}-85}{8}$$

Now solve the equation x=\frac{85±\sqrt{8185}}{-8} when ± is minus. Subtract \sqrt{8185} from 85.

$$x=\frac{85-\sqrt{8185}}{-8}$$

Divide 85-\sqrt{8185} by -8.

$$x=\frac{\sqrt{8185}-85}{8}$$

The equation is now solved.

$$x=\frac{-\sqrt{8185}-85}{8} x=\frac{\sqrt{8185}-85}{8}$$

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 20x, the least common multiple of -5,x,4.

$$-4xx+20\times 3=5x\left(4\times 4+1\right)$$

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

$$-4x^{2}+20\times 3=5x\left(4\times 4+1\right)$$

Multiply 20 and 3 to get 60.

$$-4x^{2}+60=5x\left(4\times 4+1\right)$$

Multiply 4 and 4 to get 16.

$$-4x^{2}+60=5x\left(16+1\right)$$

Add 16 and 1 to get 17.

$$-4x^{2}+60=5x\times 17$$

Multiply 5 and 17 to get 85.

$$-4x^{2}+60=85x$$

Subtract 85x from both sides.

$$-4x^{2}+60-85x=0$$

Subtract 60 from both sides. Anything subtracted from zero gives its negation.

$$-4x^{2}-85x=-60$$

Divide both sides by -4.

$$\frac{-4x^{2}-85x}{-4}=-\frac{60}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\left(-\frac{85}{-4}\right)x=-\frac{60}{-4}$$

Divide -85 by -4.

$$x^{2}+\frac{85}{4}x=-\frac{60}{-4}$$

Divide -60 by -4.

$$x^{2}+\frac{85}{4}x=15$$

Divide \frac{85}{4}, the coefficient of the x term, by 2 to get \frac{85}{8}. Then add the square of \frac{85}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{85}{4}x+\left(\frac{85}{8}\right)^{2}=15+\left(\frac{85}{8}\right)^{2}$$

Square \frac{85}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{85}{4}x+\frac{7225}{64}=15+\frac{7225}{64}$$

Add 15 to \frac{7225}{64}.

$$x^{2}+\frac{85}{4}x+\frac{7225}{64}=\frac{8185}{64}$$

Factor x^{2}+\frac{85}{4}x+\frac{7225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{85}{8}\right)^{2}=\frac{8185}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{85}{8}\right)^{2}}=\sqrt{\frac{8185}{64}}$$

Simplify.

$$x+\frac{85}{8}=\frac{\sqrt{8185}}{8} x+\frac{85}{8}=-\frac{\sqrt{8185}}{8}$$

Subtract \frac{85}{8} from both sides of the equation.

$$x=\frac{\sqrt{8185}-85}{8} x=\frac{-\sqrt{8185}-85}{8}$$