Solve 4/x+x/35=3/2

$$\frac { 4 } { x } + \frac { x } { 35 } = \frac { 3 } { 2 }$$

Solve for x

$$x = \frac{\sqrt{8785} + 105}{4} \approx 49.682082707$$

$$x = \frac{105 - \sqrt{8785}}{4} \approx 2.817917293$$

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 70x, the least common multiple of x,35,2.

$$70\times 4+2xx=105x$$

Multiply 70 and 4 to get 280.

$$280+2xx=105x$$

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

$$280+2x^{2}=105x$$

Subtract 105x from both sides.

$$280+2x^{2}-105x=0$$

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.

$$2x^{2}-105x+280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -105 for b, and 280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-105\right)±\sqrt{\left(-105\right)^{2}-4\times 2\times 280}}{2\times 2}$$

Square -105.

$$x=\frac{-\left(-105\right)±\sqrt{11025-4\times 2\times 280}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-105\right)±\sqrt{11025-8\times 280}}{2\times 2}$$

Multiply -8 times 280.

$$x=\frac{-\left(-105\right)±\sqrt{11025-2240}}{2\times 2}$$

Add 11025 to -2240.

$$x=\frac{-\left(-105\right)±\sqrt{8785}}{2\times 2}$$

The opposite of -105 is 105.

$$x=\frac{105±\sqrt{8785}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{105±\sqrt{8785}}{4}$$

Now solve the equation x=\frac{105±\sqrt{8785}}{4} when ± is plus. Add 105 to \sqrt{8785}.

$$x=\frac{\sqrt{8785}+105}{4}$$

Now solve the equation x=\frac{105±\sqrt{8785}}{4} when ± is minus. Subtract \sqrt{8785} from 105.

$$x=\frac{105-\sqrt{8785}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{8785}+105}{4} x=\frac{105-\sqrt{8785}}{4}$$

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 70x, the least common multiple of x,35,2.

$$70\times 4+2xx=105x$$

Multiply 70 and 4 to get 280.

$$280+2xx=105x$$

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

$$280+2x^{2}=105x$$

Subtract 105x from both sides.

$$280+2x^{2}-105x=0$$

Subtract 280 from both sides. Anything subtracted from zero gives its negation.

$$2x^{2}-105x=-280$$

Divide both sides by 2.

$$\frac{2x^{2}-105x}{2}=-\frac{280}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{105}{2}x=-\frac{280}{2}$$

Divide -280 by 2.

$$x^{2}-\frac{105}{2}x=-140$$

Divide -\frac{105}{2}, the coefficient of the x term, by 2 to get -\frac{105}{4}. Then add the square of -\frac{105}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{105}{2}x+\left(-\frac{105}{4}\right)^{2}=-140+\left(-\frac{105}{4}\right)^{2}$$

Square -\frac{105}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{105}{2}x+\frac{11025}{16}=-140+\frac{11025}{16}$$

Add -140 to \frac{11025}{16}.

$$x^{2}-\frac{105}{2}x+\frac{11025}{16}=\frac{8785}{16}$$

Factor x^{2}-\frac{105}{2}x+\frac{11025}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{105}{4}\right)^{2}=\frac{8785}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{105}{4}\right)^{2}}=\sqrt{\frac{8785}{16}}$$

Simplify.

$$x-\frac{105}{4}=\frac{\sqrt{8785}}{4} x-\frac{105}{4}=-\frac{\sqrt{8785}}{4}$$

Add \frac{105}{4} to both sides of the equation.

$$x=\frac{\sqrt{8785}+105}{4} x=\frac{105-\sqrt{8785}}{4}$$