Solve 49.7=4x^2/(1-x)(2-x)

$$49.7= \frac{ 4 { x }^{ 2 } }{ (1-x)(2-x) }$$

Solve for x

$$x = \frac{\sqrt{406049} + 1491}{914} \approx 2.328467989$$

$$x=\frac{1491-\sqrt{406049}}{914}\approx 0.934114067$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right).

$$49.7\left(x-2\right)\left(x-1\right)=4x^{2}$$

Use the distributive property to multiply 49.7 by x-2.

$$\left(49.7x-99.4\right)\left(x-1\right)=4x^{2}$$

Use the distributive property to multiply 49.7x-99.4 by x-1 and combine like terms.

$$49.7x^{2}-149.1x+99.4=4x^{2}$$

Subtract 4x^{2} from both sides.

$$49.7x^{2}-149.1x+99.4-4x^{2}=0$$

Combine 49.7x^{2} and -4x^{2} to get 45.7x^{2}.

$$45.7x^{2}-149.1x+99.4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 45.7 for a, -149.1 for b, and 99.4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-149.1\right)±\sqrt{\left(-149.1\right)^{2}-4\times 45.7\times 99.4}}{2\times 45.7}$$

Square -149.1 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-149.1\right)±\sqrt{22230.81-4\times 45.7\times 99.4}}{2\times 45.7}$$

Multiply -4 times 45.7.

$$x=\frac{-\left(-149.1\right)±\sqrt{22230.81-182.8\times 99.4}}{2\times 45.7}$$

Multiply -182.8 times 99.4 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-149.1\right)±\sqrt{22230.81-18170.32}}{2\times 45.7}$$

Add 22230.81 to -18170.32 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-149.1\right)±\sqrt{4060.49}}{2\times 45.7}$$

Take the square root of 4060.49.

$$x=\frac{-\left(-149.1\right)±\frac{\sqrt{406049}}{10}}{2\times 45.7}$$

The opposite of -149.1 is 149.1.

$$x=\frac{149.1±\frac{\sqrt{406049}}{10}}{2\times 45.7}$$

Multiply 2 times 45.7.

$$x=\frac{149.1±\frac{\sqrt{406049}}{10}}{91.4}$$

Now solve the equation x=\frac{149.1±\frac{\sqrt{406049}}{10}}{91.4} when ± is plus. Add 149.1 to \frac{\sqrt{406049}}{10}.

$$x=\frac{\sqrt{406049}+1491}{10\times 91.4}$$

Divide \frac{1491+\sqrt{406049}}{10} by 91.4 by multiplying \frac{1491+\sqrt{406049}}{10} by the reciprocal of 91.4.

$$x=\frac{\sqrt{406049}+1491}{914}$$

Now solve the equation x=\frac{149.1±\frac{\sqrt{406049}}{10}}{91.4} when ± is minus. Subtract \frac{\sqrt{406049}}{10} from 149.1.

$$x=\frac{1491-\sqrt{406049}}{10\times 91.4}$$

Divide \frac{1491-\sqrt{406049}}{10} by 91.4 by multiplying \frac{1491-\sqrt{406049}}{10} by the reciprocal of 91.4.

$$x=\frac{1491-\sqrt{406049}}{914}$$

The equation is now solved.

$$x=\frac{\sqrt{406049}+1491}{914} x=\frac{1491-\sqrt{406049}}{914}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 1,2 since division by zero is not defined. Multiply both sides of the equation by \left(x-2\right)\left(x-1\right).

$$49.7\left(x-2\right)\left(x-1\right)=4x^{2}$$

Use the distributive property to multiply 49.7 by x-2.

$$\left(49.7x-99.4\right)\left(x-1\right)=4x^{2}$$

Use the distributive property to multiply 49.7x-99.4 by x-1 and combine like terms.

$$49.7x^{2}-149.1x+99.4=4x^{2}$$

Subtract 4x^{2} from both sides.

$$49.7x^{2}-149.1x+99.4-4x^{2}=0$$

Combine 49.7x^{2} and -4x^{2} to get 45.7x^{2}.

$$45.7x^{2}-149.1x+99.4=0$$

Subtract 99.4 from both sides. Anything subtracted from zero gives its negation.

$$45.7x^{2}-149.1x=-99.4$$

Divide both sides of the equation by 45.7, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{45.7x^{2}-149.1x}{45.7}=-\frac{99.4}{45.7}$$

Dividing by 45.7 undoes the multiplication by 45.7.

$$x^{2}+\left(-\frac{149.1}{45.7}\right)x=-\frac{99.4}{45.7}$$

Divide -149.1 by 45.7 by multiplying -149.1 by the reciprocal of 45.7.

$$x^{2}-\frac{1491}{457}x=-\frac{99.4}{45.7}$$

Divide -99.4 by 45.7 by multiplying -99.4 by the reciprocal of 45.7.

$$x^{2}-\frac{1491}{457}x=-\frac{994}{457}$$

Divide -\frac{1491}{457}, the coefficient of the x term, by 2 to get -\frac{1491}{914}. Then add the square of -\frac{1491}{914} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1491}{457}x+\left(-\frac{1491}{914}\right)^{2}=-\frac{994}{457}+\left(-\frac{1491}{914}\right)^{2}$$

Square -\frac{1491}{914} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1491}{457}x+\frac{2223081}{835396}=-\frac{994}{457}+\frac{2223081}{835396}$$

Add -\frac{994}{457} to \frac{2223081}{835396} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{1491}{457}x+\frac{2223081}{835396}=\frac{406049}{835396}$$

Factor x^{2}-\frac{1491}{457}x+\frac{2223081}{835396}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1491}{914}\right)^{2}=\frac{406049}{835396}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1491}{914}\right)^{2}}=\sqrt{\frac{406049}{835396}}$$

Simplify.

$$x-\frac{1491}{914}=\frac{\sqrt{406049}}{914} x-\frac{1491}{914}=-\frac{\sqrt{406049}}{914}$$

Add \frac{1491}{914} to both sides of the equation.

$$x=\frac{\sqrt{406049}+1491}{914} x=\frac{1491-\sqrt{406049}}{914}$$