Solve x^2/(4-x)(1-x)=13*10^9

$$\frac { x ^ { 2 } } { ( 4 - x ) ( 1 - x ) } = 13 \times 10 ^ { 9 }$$

Solve for x

$$x = \frac{20000 \sqrt{950625000130} + 32500000000}{12999999999} \approx 4$$

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}\approx 1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-1\right).

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Calculate 10 to the power of 9 and get 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Multiply 13 and 1000000000 to get 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Use the distributive property to multiply 13000000000 by x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Use the distributive property to multiply 13000000000x-52000000000 by x-1 and combine like terms.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Subtract 13000000000x^{2} from both sides.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Combine x^{2} and -13000000000x^{2} to get -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Add 65000000000x to both sides.

$$-12999999999x^{2}[phone]x=52000000000$$

Subtract 52000000000 from both sides.

$$-12999999999x^{2}[phone]x-52000000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12999999999 for a, 65000000000 for b, and -52000000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65000000000±\sqrt{65000000000^{2}-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Square 65000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Multiply -4 times -12999999999.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000+51999999996\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Multiply 51999999996 times -52000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-2703999999792000000000}}{2\left(-12999999999\right)}$$

Add 4225000000000000000000 to -2703999999792000000000.

$$x=\frac{-65000000000±\sqrt{1521000000208000000000}}{2\left(-12999999999\right)}$$

Take the square root of 1521000000208000000000.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{2\left(-12999999999\right)}$$

Multiply 2 times -12999999999.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}$$

Now solve the equation x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} when ± is plus. Add -65000000000 to 40000\sqrt{950625000130}.

$$x=\frac{40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Divide -65000000000+40000\sqrt{950625000130} by -25999999998.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$

Now solve the equation x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} when ± is minus. Subtract 40000\sqrt{950625000130} from -65000000000.

$$x=\frac{-40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Divide -65000000000-40000\sqrt{950625000130} by -25999999998.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

The equation is now solved.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999} x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,4 since division by zero is not defined. Multiply both sides of the equation by \left(x-4\right)\left(x-1\right).

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Calculate 10 to the power of 9 and get 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Multiply 13 and 1000000000 to get 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Use the distributive property to multiply 13000000000 by x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Use the distributive property to multiply 13000000000x-52000000000 by x-1 and combine like terms.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Subtract 13000000000x^{2} from both sides.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Combine x^{2} and -13000000000x^{2} to get -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Add 65000000000x to both sides.

$$-12999999999x^{2}[phone]x=52000000000$$

Divide both sides by -12999999999.

$$\frac{-12999999999x^{2}[phone]x}{-12999999999}=\frac{52000000000}{-12999999999}$$

Dividing by -12999999999 undoes the multiplication by -12999999999.

$$x^{2}+\frac{65000000000}{-12999999999}x=\frac{52000000000}{-12999999999}$$

Divide 65000000000 by -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=\frac{52000000000}{-12999999999}$$

Divide 52000000000 by -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=-\frac{52000000000}{12999999999}$$

Divide -\frac{65000000000}{12999999999}, the coefficient of the x term, by 2 to get -\frac{32500000000}{12999999999}. Then add the square of -\frac{32500000000}{12999999999} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{65000000000}{12999999999}x+\left(-\frac{32500000000}{12999999999}\right)^{2}=-\frac{52000000000}{12999999999}+\left(-\frac{32500000000}{12999999999}\right)^{2}$$

Square -\frac{32500000000}{12999999999} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=-\frac{52000000000}{12999999999}+\frac{1056250000000000000000}{168999999974000000001}$$

Add -\frac{52000000000}{12999999999} to \frac{1056250000000000000000}{168999999974000000001} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=\frac{380250000052000000000}{168999999974000000001}$$

Factor x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{32500000000}{12999999999}\right)^{2}=\frac{380250000052000000000}{168999999974000000001}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{32500000000}{12999999999}\right)^{2}}=\sqrt{\frac{380250000052000000000}{168999999974000000001}}$$

Simplify.

$$x-\frac{32500000000}{12999999999}=\frac{20000\sqrt{950625000130}}{12999999999} x-\frac{32500000000}{12999999999}=-\frac{20000\sqrt{950625000130}}{12999999999}$$

Add \frac{32500000000}{12999999999} to both sides of the equation.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999} x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$