Solve 70/x+35+70/x-35=40

$$\frac{ 70 }{ x+35 } + \frac{ 70 }{ x-35 } =40$$

Solve for x

$$x = \frac{7 \sqrt{401} + 7}{4} \approx 36.79372269$$

$$x=\frac{7-7\sqrt{401}}{4}\approx -33.29372269$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -35,35 since division by zero is not defined. Multiply both sides of the equation by \left(x-35\right)\left(x+35\right), the least common multiple of x+35,x-35.

$$\left(x-35\right)\times 70+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Use the distributive property to multiply x-35 by 70.

$$70x-2450+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Use the distributive property to multiply x+35 by 70.

$$70x-2450+70x+2450=40\left(x-35\right)\left(x+35\right)$$

Combine 70x and 70x to get 140x.

$$140x-2450+2450=40\left(x-35\right)\left(x+35\right)$$

Add -2450 and 2450 to get 0.

$$140x=40\left(x-35\right)\left(x+35\right)$$

Use the distributive property to multiply 40 by x-35.

$$140x=\left(40x-1400\right)\left(x+35\right)$$

Use the distributive property to multiply 40x-1400 by x+35 and combine like terms.

$$140x=40x^{2}-49000$$

Subtract 40x^{2} from both sides.

$$140x-40x^{2}=-49000$$

Add 49000 to both sides.

$$140x-40x^{2}+49000=0$$

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.

$$-40x^{2}+140x+49000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -40 for a, 140 for b, and 49000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140±\sqrt{140^{2}-4\left(-40\right)\times 49000}}{2\left(-40\right)}$$

Square 140.

$$x=\frac{-140±\sqrt{19600-4\left(-40\right)\times 49000}}{2\left(-40\right)}$$

Multiply -4 times -40.

$$x=\frac{-140±\sqrt{19600+160\times 49000}}{2\left(-40\right)}$$

Multiply 160 times 49000.

$$x=\frac{-140±\sqrt{19600+7840000}}{2\left(-40\right)}$$

Add 19600 to 7840000.

$$x=\frac{-140±\sqrt{7859600}}{2\left(-40\right)}$$

Take the square root of 7859600.

$$x=\frac{-140±140\sqrt{401}}{2\left(-40\right)}$$

Multiply 2 times -40.

$$x=\frac{-140±140\sqrt{401}}{-80}$$

Now solve the equation x=\frac{-140±140\sqrt{401}}{-80} when ± is plus. Add -140 to 140\sqrt{401}.

$$x=\frac{140\sqrt{401}-140}{-80}$$

Divide -140+140\sqrt{401} by -80.

$$x=\frac{7-7\sqrt{401}}{4}$$

Now solve the equation x=\frac{-140±140\sqrt{401}}{-80} when ± is minus. Subtract 140\sqrt{401} from -140.

$$x=\frac{-140\sqrt{401}-140}{-80}$$

Divide -140-140\sqrt{401} by -80.

$$x=\frac{7\sqrt{401}+7}{4}$$

The equation is now solved.

$$x=\frac{7-7\sqrt{401}}{4} x=\frac{7\sqrt{401}+7}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -35,35 since division by zero is not defined. Multiply both sides of the equation by \left(x-35\right)\left(x+35\right), the least common multiple of x+35,x-35.

$$\left(x-35\right)\times 70+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Use the distributive property to multiply x-35 by 70.

$$70x-2450+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

Use the distributive property to multiply x+35 by 70.

$$70x-2450+70x+2450=40\left(x-35\right)\left(x+35\right)$$

Combine 70x and 70x to get 140x.

$$140x-2450+2450=40\left(x-35\right)\left(x+35\right)$$

Add -2450 and 2450 to get 0.

$$140x=40\left(x-35\right)\left(x+35\right)$$

Use the distributive property to multiply 40 by x-35.

$$140x=\left(40x-1400\right)\left(x+35\right)$$

Use the distributive property to multiply 40x-1400 by x+35 and combine like terms.

$$140x=40x^{2}-49000$$

Subtract 40x^{2} from both sides.

$$140x-40x^{2}=-49000$$

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.

$$-40x^{2}+140x=-49000$$

Divide both sides by -40.

$$\frac{-40x^{2}+140x}{-40}=-\frac{49000}{-40}$$

Dividing by -40 undoes the multiplication by -40.

$$x^{2}+\frac{140}{-40}x=-\frac{49000}{-40}$$

Reduce the fraction \frac{140}{-40} to lowest terms by extracting and canceling out 20.

$$x^{2}-\frac{7}{2}x=-\frac{49000}{-40}$$

Divide -49000 by -40.

$$x^{2}-\frac{7}{2}x=1225$$

Divide -\frac{7}{2}, the coefficient of the x term, by 2 to get -\frac{7}{4}. Then add the square of -\frac{7}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=1225+\left(-\frac{7}{4}\right)^{2}$$

Square -\frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=1225+\frac{49}{16}$$

Add 1225 to \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{19649}{16}$$

Factor x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{19649}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{19649}{16}}$$

Simplify.

$$x-\frac{7}{4}=\frac{7\sqrt{401}}{4} x-\frac{7}{4}=-\frac{7\sqrt{401}}{4}$$

Add \frac{7}{4} to both sides of the equation.

$$x=\frac{7\sqrt{401}+7}{4} x=\frac{7-7\sqrt{401}}{4}$$