Solve 9x/490-x=16left(490-xright)/x

$$\frac{ 9x }{ 490-x } = \frac{ 16 \left( 490-x \right) }{ x }$$

Solve for x

$$x=1960$$

$$x=280$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 0,490 since division by zero is not defined. Multiply both sides of the equation by x\left(x-490\right), the least common multiple of 490-x,x.

$$-x\times 9x=\left(x-490\right)\times 16\left(490-x\right)$$

Multiply -1 and 9 to get -9.

$$-9xx=\left(x-490\right)\times 16\left(490-x\right)$$

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

$$-9x^{2}=\left(x-490\right)\times 16\left(490-x\right)$$

Use the distributive property to multiply x-490 by 16.

$$-9x^{2}=\left(16x-7840\right)\left(490-x\right)$$

Use the distributive property to multiply 16x-7840 by 490-x and combine like terms.

$$-9x^{2}=15680x-16x^{2}-3841600$$

Subtract 15680x from both sides.

$$-9x^{2}-15680x=-16x^{2}-3841600$$

Add 16x^{2} to both sides.

$$-9x^{2}-15680x+16x^{2}=-3841600$$

Combine -9x^{2} and 16x^{2} to get 7x^{2}.

$$7x^{2}-15680x=-3841600$$

Add 3841600 to both sides.

$$7x^{2}-15680x+3841600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -15680 for b, and 3841600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15680\right)±\sqrt{\left(-15680\right)^{2}-4\times 7\times 3841600}}{2\times 7}$$

Square -15680.

$$x=\frac{-\left(-15680\right)±\sqrt{245862400-4\times 7\times 3841600}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-15680\right)±\sqrt{245862400-28\times 3841600}}{2\times 7}$$

Multiply -28 times 3841600.

$$x=\frac{-\left(-15680\right)±\sqrt{245862400-107564800}}{2\times 7}$$

Add 245862400 to -107564800.

$$x=\frac{-\left(-15680\right)±\sqrt{138297600}}{2\times 7}$$

Take the square root of 138297600.

$$x=\frac{-\left(-15680\right)±11760}{2\times 7}$$

The opposite of -15680 is 15680.

$$x=\frac{15680±11760}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{15680±11760}{14}$$

Now solve the equation x=\frac{15680±11760}{14} when ± is plus. Add 15680 to 11760.

$$x=\frac{27440}{14}$$

Divide 27440 by 14.

$$x=1960$$

Now solve the equation x=\frac{15680±11760}{14} when ± is minus. Subtract 11760 from 15680.

$$x=\frac{3920}{14}$$

Divide 3920 by 14.

$$x=280$$

The equation is now solved.

$$x=1960 x=280$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 0,490 since division by zero is not defined. Multiply both sides of the equation by x\left(x-490\right), the least common multiple of 490-x,x.

$$-x\times 9x=\left(x-490\right)\times 16\left(490-x\right)$$

Multiply -1 and 9 to get -9.

$$-9xx=\left(x-490\right)\times 16\left(490-x\right)$$

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

$$-9x^{2}=\left(x-490\right)\times 16\left(490-x\right)$$

Use the distributive property to multiply x-490 by 16.

$$-9x^{2}=\left(16x-7840\right)\left(490-x\right)$$

Use the distributive property to multiply 16x-7840 by 490-x and combine like terms.

$$-9x^{2}=15680x-16x^{2}-3841600$$

Subtract 15680x from both sides.

$$-9x^{2}-15680x=-16x^{2}-3841600$$

Add 16x^{2} to both sides.

$$-9x^{2}-15680x+16x^{2}=-3841600$$

Combine -9x^{2} and 16x^{2} to get 7x^{2}.

$$7x^{2}-15680x=-3841600$$

Divide both sides by 7.

$$\frac{7x^{2}-15680x}{7}=-\frac{3841600}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}+\left(-\frac{15680}{7}\right)x=-\frac{3841600}{7}$$

Divide -15680 by 7.

$$x^{2}-2240x=-\frac{3841600}{7}$$

Divide -3841600 by 7.

$$x^{2}-2240x=-548800$$

Divide -2240, the coefficient of the x term, by 2 to get -1120. Then add the square of -1120 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2240x+\left(-1120\right)^{2}=-548800+\left(-1120\right)^{2}$$

Square -1120.

$$x^{2}-2240x+1254400=-548800+1254400$$

Add -548800 to 1254400.

$$x^{2}-2240x+1254400=705600$$

Factor x^{2}-2240x+1254400. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1120\right)^{2}=705600$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1120\right)^{2}}=\sqrt{705600}$$

Simplify.

$$x-1120=840 x-1120=-840$$

Add 1120 to both sides of the equation.

$$x=1960 x=280$$