Solve 2200/100-x+15=22*100/67-x

$$\frac{ 2200 }{ 100-x } +15= \frac{ 22 \times 100 }{ 67-x }$$

Solve for x

$$x=12$$

$$x=155$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 67,100 since division by zero is not defined. Multiply both sides of the equation by \left(x-100\right)\left(x-67\right), the least common multiple of 100-x,67-x.

$$\left(67-x\right)\times 2200+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Use the distributive property to multiply 67-x by 2200.

$$147400-2200x+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Use the distributive property to multiply x-100 by x-67 and combine like terms.

$$147400-2200x+\left(x^{2}-167x+6700\right)\times 15=\left(100-x\right)\times 22\times 100$$

Use the distributive property to multiply x^{2}-167x+6700 by 15.

$$147400-2200x+15x^{2}-2505x+100500=\left(100-x\right)\times 22\times 100$$

Combine -2200x and -2505x to get -4705x.

$$147400-4705x+15x^{2}+100500=\left(100-x\right)\times 22\times 100$$

Add 147400 and 100500 to get 247900.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 22\times 100$$

Multiply 22 and 100 to get 2200.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 2200$$

Use the distributive property to multiply 100-x by 2200.

$$247900-4705x+15x^{2}=220000-2200x$$

Subtract 220000 from both sides.

$$247900-4705x+15x^{2}-220000=-2200x$$

Subtract 220000 from 247900 to get 27900.

$$27900-4705x+15x^{2}=-2200x$$

Add 2200x to both sides.

$$27900-4705x+15x^{2}+2200x=0$$

Combine -4705x and 2200x to get -2505x.

$$27900-2505x+15x^{2}=0$$

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.

$$15x^{2}-2505x+27900=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -2505 for b, and 27900 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2505\right)±\sqrt{\left(-2505\right)^{2}-4\times 15\times 27900}}{2\times 15}$$

Square -2505.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-4\times 15\times 27900}}{2\times 15}$$

Multiply -4 times 15.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-60\times 27900}}{2\times 15}$$

Multiply -60 times 27900.

$$x=\frac{-\left(-2505\right)±\sqrt{6275025-1674000}}{2\times 15}$$

Add 6275025 to -1674000.

$$x=\frac{-\left(-2505\right)±\sqrt{4601025}}{2\times 15}$$

Take the square root of 4601025.

$$x=\frac{-\left(-2505\right)±2145}{2\times 15}$$

The opposite of -2505 is 2505.

$$x=\frac{2505±2145}{2\times 15}$$

Multiply 2 times 15.

$$x=\frac{2505±2145}{30}$$

Now solve the equation x=\frac{2505±2145}{30} when ± is plus. Add 2505 to 2145.

$$x=\frac{4650}{30}$$

Divide 4650 by 30.

$$x=155$$

Now solve the equation x=\frac{2505±2145}{30} when ± is minus. Subtract 2145 from 2505.

$$x=\frac{360}{30}$$

Divide 360 by 30.

$$x=12$$

The equation is now solved.

$$x=155 x=12$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 67,100 since division by zero is not defined. Multiply both sides of the equation by \left(x-100\right)\left(x-67\right), the least common multiple of 100-x,67-x.

$$\left(67-x\right)\times 2200+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Use the distributive property to multiply 67-x by 2200.

$$147400-2200x+\left(x-100\right)\left(x-67\right)\times 15=\left(100-x\right)\times 22\times 100$$

Use the distributive property to multiply x-100 by x-67 and combine like terms.

$$147400-2200x+\left(x^{2}-167x+6700\right)\times 15=\left(100-x\right)\times 22\times 100$$

Use the distributive property to multiply x^{2}-167x+6700 by 15.

$$147400-2200x+15x^{2}-2505x+100500=\left(100-x\right)\times 22\times 100$$

Combine -2200x and -2505x to get -4705x.

$$147400-4705x+15x^{2}+100500=\left(100-x\right)\times 22\times 100$$

Add 147400 and 100500 to get 247900.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 22\times 100$$

Multiply 22 and 100 to get 2200.

$$247900-4705x+15x^{2}=\left(100-x\right)\times 2200$$

Use the distributive property to multiply 100-x by 2200.

$$247900-4705x+15x^{2}=220000-2200x$$

Add 2200x to both sides.

$$247900-4705x+15x^{2}+2200x=220000$$

Combine -4705x and 2200x to get -2505x.

$$247900-2505x+15x^{2}=220000$$

Subtract 247900 from both sides.

$$-2505x+15x^{2}=220000-247900$$

Subtract 247900 from 220000 to get -27900.

$$-2505x+15x^{2}=-27900$$

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.

$$15x^{2}-2505x=-27900$$

Divide both sides by 15.

$$\frac{15x^{2}-2505x}{15}=-\frac{27900}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$x^{2}+\left(-\frac{2505}{15}\right)x=-\frac{27900}{15}$$

Divide -2505 by 15.

$$x^{2}-167x=-\frac{27900}{15}$$

Divide -27900 by 15.

$$x^{2}-167x=-1860$$

Divide -167, the coefficient of the x term, by 2 to get -\frac{167}{2}. Then add the square of -\frac{167}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-167x+\left(-\frac{167}{2}\right)^{2}=-1860+\left(-\frac{167}{2}\right)^{2}$$

Square -\frac{167}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-167x+\frac{27889}{4}=-1860+\frac{27889}{4}$$

Add -1860 to \frac{27889}{4}.

$$x^{2}-167x+\frac{27889}{4}=\frac{20449}{4}$$

Factor x^{2}-167x+\frac{27889}{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}.

$$\left(x-\frac{167}{2}\right)^{2}=\frac{20449}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{167}{2}\right)^{2}}=\sqrt{\frac{20449}{4}}$$

Simplify.

$$x-\frac{167}{2}=\frac{143}{2} x-\frac{167}{2}=-\frac{143}{2}$$

Add \frac{167}{2} to both sides of the equation.

$$x=155 x=12$$