Solve 6500/x+15=6500/x-30

$$\frac{ 6500 }{ x+15 } = \frac{ 6500 }{ x } -30$$

Solve for x

$$x=-65$$

$$x=50$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+15\right), the least common multiple of x+15,x.

$$x\times 6500=\left(x+15\right)\times 6500+x\left(x+15\right)\left(-30\right)$$

Use the distributive property to multiply x+15 by 6500.

$$x\times 6500=6500x+97500+x\left(x+15\right)\left(-30\right)$$

Use the distributive property to multiply x by x+15.

$$x\times 6500=6500x+97500+\left(x^{2}+15x\right)\left(-30\right)$$

Use the distributive property to multiply x^{2}+15x by -30.

$$x\times 6500=6500x+97500-30x^{2}-450x$$

Combine 6500x and -450x to get 6050x.

$$x\times 6500=6050x+97500-30x^{2}$$

Subtract 6050x from both sides.

$$x\times 6500-6050x=97500-30x^{2}$$

Combine x\times 6500 and -6050x to get 450x.

$$450x=97500-30x^{2}$$

Subtract 97500 from both sides.

$$450x-97500=-30x^{2}$$

Add 30x^{2} to both sides.

$$450x-97500+30x^{2}=0$$

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.

$$30x^{2}+450x-97500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 30 for a, 450 for b, and -97500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-450±\sqrt{450^{2}-4\times 30\left(-97500\right)}}{2\times 30}$$

Square 450.

$$x=\frac{-450±\sqrt{202500-4\times 30\left(-97500\right)}}{2\times 30}$$

Multiply -4 times 30.

$$x=\frac{-450±\sqrt{202500-120\left(-97500\right)}}{2\times 30}$$

Multiply -120 times -97500.

$$x=\frac{-450±\sqrt{202500+11700000}}{2\times 30}$$

Add 202500 to 11700000.

$$x=\frac{-450±\sqrt{11902500}}{2\times 30}$$

Take the square root of 11902500.

$$x=\frac{-450±3450}{2\times 30}$$

Multiply 2 times 30.

$$x=\frac{-450±3450}{60}$$

Now solve the equation x=\frac{-450±3450}{60} when ± is plus. Add -450 to 3450.

$$x=\frac{3000}{60}$$

Divide 3000 by 60.

$$x=50$$

Now solve the equation x=\frac{-450±3450}{60} when ± is minus. Subtract 3450 from -450.

$$x=-\frac{3900}{60}$$

Divide -3900 by 60.

$$x=-65$$

The equation is now solved.

$$x=50 x=-65$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -15,0 since division by zero is not defined. Multiply both sides of the equation by x\left(x+15\right), the least common multiple of x+15,x.

$$x\times 6500=\left(x+15\right)\times 6500+x\left(x+15\right)\left(-30\right)$$

Use the distributive property to multiply x+15 by 6500.

$$x\times 6500=6500x+97500+x\left(x+15\right)\left(-30\right)$$

Use the distributive property to multiply x by x+15.

$$x\times 6500=6500x+97500+\left(x^{2}+15x\right)\left(-30\right)$$

Use the distributive property to multiply x^{2}+15x by -30.

$$x\times 6500=6500x+97500-30x^{2}-450x$$

Combine 6500x and -450x to get 6050x.

$$x\times 6500=6050x+97500-30x^{2}$$

Subtract 6050x from both sides.

$$x\times 6500-6050x=97500-30x^{2}$$

Combine x\times 6500 and -6050x to get 450x.

$$450x=97500-30x^{2}$$

Add 30x^{2} to both sides.

$$450x+30x^{2}=97500$$

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.

$$30x^{2}+450x=97500$$

Divide both sides by 30.

$$\frac{30x^{2}+450x}{30}=\frac{97500}{30}$$

Dividing by 30 undoes the multiplication by 30.

$$x^{2}+\frac{450}{30}x=\frac{97500}{30}$$

Divide 450 by 30.

$$x^{2}+15x=\frac{97500}{30}$$

Divide 97500 by 30.

$$x^{2}+15x=3250$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=3250+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=3250+\frac{225}{4}$$

Add 3250 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{13225}{4}$$

Factor x^{2}+15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{13225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{13225}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{115}{2} x+\frac{15}{2}=-\frac{115}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=50 x=-65$$