Algebra Examples

$\frac{x^{2}}{1 - x} = 1 x \cdot 10^{- 8}$

Step 1

Simplify both sides.

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Step 1.1

Multiply $x$ by $1$ .

$\frac{x^{2}}{1 - x} = x \cdot 10^{- 8}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x^{2}}{1 - x} = x \cdot \frac{1}{10^{8}}$

Step 1.3

Raise $10$ to the power of $8$ .

$\frac{x^{2}}{1 - x} = x \cdot \frac{1}{100000000}$

Step 1.4

Combine $x$ and $\frac{1}{100000000}$ .

$\frac{x^{2}}{1 - x} = \frac{x}{100000000}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$x^{2} \cdot 100000000 = (1 - x) x$

Step 3

Solve the equation for $x$ .

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Step 3.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$(1 - x) x = x^{2} \cdot 100000000$

Step 3.2

Simplify $(1 - x) x$ .

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Step 3.2.1

Rewrite.

$0 + 0 + (1 - x) x = x^{2} \cdot 100000000$

Step 3.2.2

Simplify by adding zeros.

$(1 - x) x = x^{2} \cdot 100000000$

Step 3.2.3

Apply the distributive property.

$1 x - x \cdot x = x^{2} \cdot 100000000$

Step 3.2.4

Multiply $x$ by $1$ .

$x - x \cdot x = x^{2} \cdot 100000000$

Step 3.2.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.5.1

Move $x$ .

$x - (x \cdot x) = x^{2} \cdot 100000000$

Step 3.2.5.2

Multiply $x$ by $x$ .

$x - x^{2} = x^{2} \cdot 100000000$

Step 3.3

Move $100000000$ to the left of $x^{2}$ .

$x - x^{2} = 100000000 x^{2}$

Step 3.4

Move all terms containing $x$ to the left side of the equation.

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Step 3.4.1

Subtract $100000000 x^{2}$ from both sides of the equation.

$x - x^{2} - 100000000 x^{2} = 0$

Step 3.4.2

Subtract $100000000 x^{2}$ from $- x^{2}$ .

$x - 100000001 x^{2} = 0$

Step 3.5

Factor the left side of the equation.

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Step 3.5.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$u - 100000001 u^{2} = 0$

Step 3.5.2

Factor $u$ out of $u - 100000001 u^{2}$ .

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Step 3.5.2.1

Raise $u$ to the power of $1$ .

$u - 100000001 u^{2} = 0$

Step 3.5.2.2

Factor $u$ out of $u^{1}$ .

$u \cdot 1 - 100000001 u^{2} = 0$

Step 3.5.2.3

Factor $u$ out of $- 100000001 u^{2}$ .

$u \cdot 1 + u (- 100000001 u) = 0$

Step 3.5.2.4

Factor $u$ out of $u \cdot 1 + u (- 100000001 u)$ .

$u (1 - 100000001 u) = 0$

Step 3.5.3

Replace all occurrences of $u$ with $x$ .

$x (1 - 100000001 x) = 0$

Step 3.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 - 100000001 x = 0$

Step 3.7

Set $x$ equal to $0$ .

$x = 0$

Step 3.8

Set $1 - 100000001 x$ equal to $0$ and solve for $x$ .

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Step 3.8.1

Set $1 - 100000001 x$ equal to $0$ .

$1 - 100000001 x = 0$

Step 3.8.2

Solve $1 - 100000001 x = 0$ for $x$ .

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Step 3.8.2.1

Subtract $1$ from both sides of the equation.

$- 100000001 x = - 1$

Step 3.8.2.2

Divide each term in $- 100000001 x = - 1$ by $- 100000001$ and simplify.

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Step 3.8.2.2.1

Divide each term in $- 100000001 x = - 1$ by $- 100000001$ .

$\frac{- 100000001 x}{- 100000001} = \frac{- 1}{- 100000001}$

Step 3.8.2.2.2

Simplify the left side.

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Step 3.8.2.2.2.1

Cancel the common factor of $- 100000001$ .

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Step 3.8.2.2.2.1.1

Cancel the common factor.

$\frac{- 100000001 x}{- 100000001} = \frac{- 1}{- 100000001}$

Step 3.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 100000001}$

Step 3.8.2.2.3

Simplify the right side.

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Step 3.8.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{100000001}$

Step 3.9

The final solution is all the values that make $x (1 - 100000001 x) = 0$ true.

$x = 0, \frac{1}{100000001}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 0, \frac{1}{100000001}$

Decimal Form:

$x = 0, 9.9999999 \cdot 10^{- 9}$