Solve (1.21-x)*10+100=122.64/x

$$(1.21-x) \times 10+100= \frac{ 122.64 }{ x }$$

Solve for x

$$x = \frac{\sqrt{766081} + 1121}{200} \approx 9.981302663$$

$$x = \frac{1121 - \sqrt{766081}}{200} \approx 1.228697337$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(1.21-x\right)\times 10x+x\times 100=122.64$$

Use the distributive property to multiply 1.21-x by 10.

$$\left(12.1-10x\right)x+x\times 100=122.64$$

Use the distributive property to multiply 12.1-10x by x.

$$12.1x-10x^{2}+x\times 100=122.64$$

Combine 12.1x and x\times 100 to get 112.1x.

$$112.1x-10x^{2}=122.64$$

Subtract 122.64 from both sides.

$$112.1x-10x^{2}-122.64=0$$

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.

$$-10x^{2}+112.1x-122.64=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 112.1 for b, and -122.64 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-112.1±\sqrt{112.1^{2}-4\left(-10\right)\left(-122.64\right)}}{2\left(-10\right)}$$

Square 112.1 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-112.1±\sqrt{12566.41-4\left(-10\right)\left(-122.64\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$x=\frac{-112.1±\sqrt{12566.41+40\left(-122.64\right)}}{2\left(-10\right)}$$

Multiply 40 times -122.64.

$$x=\frac{-112.1±\sqrt{12566.41-4905.6}}{2\left(-10\right)}$$

Add 12566.41 to -4905.6 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-112.1±\sqrt{7660.81}}{2\left(-10\right)}$$

Take the square root of 7660.81.

$$x=\frac{-112.1±\frac{\sqrt{766081}}{10}}{2\left(-10\right)}$$

Multiply 2 times -10.

$$x=\frac{-112.1±\frac{\sqrt{766081}}{10}}{-20}$$

Now solve the equation x=\frac{-112.1±\frac{\sqrt{766081}}{10}}{-20} when ± is plus. Add -112.1 to \frac{\sqrt{766081}}{10}.

$$x=\frac{\sqrt{766081}-1121}{-20\times 10}$$

Divide \frac{-1121+\sqrt{766081}}{10} by -20.

$$x=\frac{1121-\sqrt{766081}}{200}$$

Now solve the equation x=\frac{-112.1±\frac{\sqrt{766081}}{10}}{-20} when ± is minus. Subtract \frac{\sqrt{766081}}{10} from -112.1.

$$x=\frac{-\sqrt{766081}-1121}{-20\times 10}$$

Divide \frac{-1121-\sqrt{766081}}{10} by -20.

$$x=\frac{\sqrt{766081}+1121}{200}$$

The equation is now solved.

$$x=\frac{1121-\sqrt{766081}}{200} x=\frac{\sqrt{766081}+1121}{200}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$\left(1.21-x\right)\times 10x+x\times 100=122.64$$

Use the distributive property to multiply 1.21-x by 10.

$$\left(12.1-10x\right)x+x\times 100=122.64$$

Use the distributive property to multiply 12.1-10x by x.

$$12.1x-10x^{2}+x\times 100=122.64$$

Combine 12.1x and x\times 100 to get 112.1x.

$$112.1x-10x^{2}=122.64$$

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.

$$-10x^{2}+112.1x=122.64$$

Divide both sides by -10.

$$\frac{-10x^{2}+112.1x}{-10}=\frac{122.64}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$x^{2}+\frac{112.1}{-10}x=\frac{122.64}{-10}$$

Divide 112.1 by -10.

$$x^{2}-11.21x=\frac{122.64}{-10}$$

Divide 122.64 by -10.

$$x^{2}-11.21x=-12.264$$

Divide -11.21, the coefficient of the x term, by 2 to get -5.605. Then add the square of -5.605 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-11.21x+\left(-5.605\right)^{2}=-12.264+\left(-5.605\right)^{2}$$

Square -5.605 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-11.21x+31.416025=-12.264+31.416025$$

Add -12.264 to 31.416025 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-11.21x+31.416025=19.152025$$

Factor x^{2}-11.21x+31.416025. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5.605\right)^{2}=19.152025$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5.605\right)^{2}}=\sqrt{19.152025}$$

Simplify.

$$x-5.605=\frac{\sqrt{766081}}{200} x-5.605=-\frac{\sqrt{766081}}{200}$$

Add 5.605 to both sides of the equation.

$$x=\frac{\sqrt{766081}+1121}{200} x=\frac{1121-\sqrt{766081}}{200}$$