Solve x^2/(4-x)(1-x)=1.3*10^9

$$\frac { x ^ { 2 } } { ( 4 - x ) ( 1 - x ) } = 1.3 \times 10 ^ { 9 }$$

Solve for x

$$x = \frac{20000 \sqrt{9506250013} + 3250000000}{1299999999} \approx 4.000000004$$

$$x=\frac{3250000000-20000\sqrt{9506250013}}{1299999999}\approx 1$$

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).

$$x^{2}=1.3\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Calculate 10 to the power of 9 and get 1000000000.

$$x^{2}=1.3\times 1000000000\left(x-4\right)\left(x-1\right)$$

Multiply 1.3 and 1000000000 to get 1300000000.

$$x^{2}=1300000000\left(x-4\right)\left(x-1\right)$$

Use the distributive property to multiply 1300000000 by x-4.

$$x^{2}=\left(1300000000x-5200000000\right)\left(x-1\right)$$

Use the distributive property to multiply 1300000000x-5200000000 by x-1 and combine like terms.

$$x^{2}=1300000000x^{2}-6500000000x+5200000000$$

Subtract 1300000000x^{2} from both sides.

$$x^{2}-1300000000x^{2}=-6500000000x+5200000000$$

Combine x^{2} and -1300000000x^{2} to get -1299999999x^{2}.

$$-1299999999x^{2}=-6500000000x+5200000000$$

Add 6500000000x to both sides.

$$-1299999999x^{2}[phone]x=5200000000$$

Subtract 5200000000 from both sides.

$$-1299999999x^{2}[phone]x-5200000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1299999999 for a, 6500000000 for b, and -5200000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6500000000±\sqrt{6500000000^{2}-4\left(-1299999999\right)\left(-5200000000\right)}}{2\left(-1299999999\right)}$$

Square 6500000000.

$$x=\frac{-6500000000±\sqrt{42250000000000000000-4\left(-1299999999\right)\left(-5200000000\right)}}{2\left(-1299999999\right)}$$

Multiply -4 times -1299999999.

$$x=\frac{-6500000000±\sqrt{42250000000000000000+5199999996\left(-5200000000\right)}}{2\left(-1299999999\right)}$$

Multiply 5199999996 times -5200000000.

$$x=\frac{-6500000000±\sqrt{42250000000000000000-27039999979200000000}}{2\left(-1299999999\right)}$$

Add 42250000000000000000 to -27039999979200000000.

$$x=\frac{-6500000000±\sqrt{15210000020800000000}}{2\left(-1299999999\right)}$$

Take the square root of 15210000020800000000.

$$x=\frac{-6500000000±40000\sqrt{9506250013}}{2\left(-1299999999\right)}$$

Multiply 2 times -1299999999.

$$x=\frac{-6500000000±40000\sqrt{9506250013}}{-2599999998}$$

Now solve the equation x=\frac{-6500000000±40000\sqrt{9506250013}}{-2599999998} when ± is plus. Add -6500000000 to 40000\sqrt{9506250013}.

$$x=\frac{40000\sqrt{9506250013}-6500000000}{-2599999998}$$

Divide -6500000000+40000\sqrt{9506250013} by -2599999998.

$$x=\frac{3250000000-20000\sqrt{9506250013}}{1299999999}$$

Now solve the equation x=\frac{-6500000000±40000\sqrt{9506250013}}{-2599999998} when ± is minus. Subtract 40000\sqrt{9506250013} from -6500000000.

$$x=\frac{-40000\sqrt{9506250013}-6500000000}{-2599999998}$$

Divide -6500000000-40000\sqrt{9506250013} by -2599999998.

$$x=\frac{20000\sqrt{9506250013}[phone]}{1299999999}$$

The equation is now solved.

$$x=\frac{3250000000-20000\sqrt{9506250013}}{1299999999} x=\frac{20000\sqrt{9506250013}[phone]}{1299999999}$$

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).

$$x^{2}=1.3\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Calculate 10 to the power of 9 and get 1000000000.

$$x^{2}=1.3\times 1000000000\left(x-4\right)\left(x-1\right)$$

Multiply 1.3 and 1000000000 to get 1300000000.

$$x^{2}=1300000000\left(x-4\right)\left(x-1\right)$$

Use the distributive property to multiply 1300000000 by x-4.

$$x^{2}=\left(1300000000x-5200000000\right)\left(x-1\right)$$

Use the distributive property to multiply 1300000000x-5200000000 by x-1 and combine like terms.

$$x^{2}=1300000000x^{2}-6500000000x+5200000000$$

Subtract 1300000000x^{2} from both sides.

$$x^{2}-1300000000x^{2}=-6500000000x+5200000000$$

Combine x^{2} and -1300000000x^{2} to get -1299999999x^{2}.

$$-1299999999x^{2}=-6500000000x+5200000000$$

Add 6500000000x to both sides.

$$-1299999999x^{2}[phone]x=5200000000$$

Divide both sides by -1299999999.

$$\frac{-1299999999x^{2}[phone]x}{-1299999999}=\frac{5200000000}{-1299999999}$$

Dividing by -1299999999 undoes the multiplication by -1299999999.

$$x^{2}+\frac{6500000000}{-1299999999}x=\frac{5200000000}{-1299999999}$$

Divide 6500000000 by -1299999999.

$$x^{2}-\frac{6500000000}{1299999999}x=\frac{5200000000}{-1299999999}$$

Divide 5200000000 by -1299999999.

$$x^{2}-\frac{6500000000}{1299999999}x=-\frac{5200000000}{1299999999}$$

Divide -\frac{6500000000}{1299999999}, the coefficient of the x term, by 2 to get -\frac{3250000000}{1299999999}. Then add the square of -\frac{3250000000}{1299999999} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6500000000}{1299999999}x+\left(-\frac{3250000000}{1299999999}\right)^{2}=-\frac{5200000000}{1299999999}+\left(-\frac{3250000000}{1299999999}\right)^{2}$$

Square -\frac{3250000000}{1299999999} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6500000000}{1299999999}x+\frac{10562500000000000000}{1689999997400000001}=-\frac{5200000000}{1299999999}+\frac{10562500000000000000}{1689999997400000001}$$

Add -\frac{5200000000}{1299999999} to \frac{10562500000000000000}{1689999997400000001} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6500000000}{1299999999}x+\frac{10562500000000000000}{1689999997400000001}=\frac{3802500005200000000}{1689999997400000001}$$

Factor x^{2}-\frac{6500000000}{1299999999}x+\frac{10562500000000000000}{1689999997400000001}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3250000000}{1299999999}\right)^{2}=\frac{3802500005200000000}{1689999997400000001}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3250000000}{1299999999}\right)^{2}}=\sqrt{\frac{3802500005200000000}{1689999997400000001}}$$

Simplify.

$$x-\frac{3250000000}{1299999999}=\frac{20000\sqrt{9506250013}}{1299999999} x-\frac{3250000000}{1299999999}=-\frac{20000\sqrt{9506250013}}{1299999999}$$

Add \frac{3250000000}{1299999999} to both sides of the equation.

$$x=\frac{20000\sqrt{9506250013}[phone]}{1299999999} x=\frac{3250000000-20000\sqrt{9506250013}}{1299999999}$$