Algebra Examples

$\frac{{(2 x)}^{2}}{(1 - x) (2 - x)} = 50.5$

Step 1

Factor each term.

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Step 1.1

Apply the product rule to $2 x$ .

$\frac{2^{2} x^{2}}{(1 - x) (2 - x)} = 50.5$

Step 1.2

Raise $2$ to the power of $2$ .

$\frac{4 x^{2}}{(1 - x) (2 - x)} = 50.5$

Step 1.3

Cancel the common factor of $4$ and $1 - x$ .

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Step 1.3.1

Reorder terms.

$\frac{4 x^{2}}{(- x + 1) (2 - x)} = 50.5$

Step 1.3.2

Factor $1$ out of $4 x^{2}$ .

$\frac{1 (4 x^{2})}{(- x + 1) (2 - x)} = 50.5$

Step 1.3.3

Cancel the common factors.

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Step 1.3.3.1

Factor $1$ out of $(- x + 1) (2 - x)$ .

$\frac{1 (4 x^{2})}{1 ((- x + 1) (2 - x))} = 50.5$

Step 1.3.3.2

Cancel the common factor.

$\frac{1 (4 x^{2})}{1 ((- x + 1) (2 - x))} = 50.5$

Step 1.3.3.3

Rewrite the expression.

$\frac{4 x^{2}}{(- x + 1) (2 - x)} = 50.5$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$(- x + 1) (2 - x), 1$

Step 2.2

The LCM of one and any expression is the expression.

$(- x + 1) (2 - x)$

Step 3

Multiply each term in $\frac{4 x^{2}}{(- x + 1) (2 - x)} = 50.5$ by $(- x + 1) (2 - x)$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{4 x^{2}}{(- x + 1) (2 - x)} = 50.5$ by $(- x + 1) (2 - x)$ .

$\frac{4 x^{2}}{(- x + 1) (2 - x)} ((- x + 1) (2 - x)) = 50.5 ((- x + 1) (2 - x))$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $(- x + 1) (2 - x)$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{4 x^{2}}{(- x + 1) (2 - x)} ((- x + 1) (2 - x)) = 50.5 ((- x + 1) (2 - x))$

Step 3.2.1.2

Rewrite the expression.

$4 x^{2} = 50.5 ((- x + 1) (2 - x))$

Step 3.3

Simplify the right side.

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Step 3.3.1

Expand $(- x + 1) (2 - x)$ using the FOIL Method.

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Step 3.3.1.1

Apply the distributive property.

$4 x^{2} = 50.5 (- x (2 - x) + 1 (2 - x))$

Step 3.3.1.2

Apply the distributive property.

$4 x^{2} = 50.5 (- x \cdot 2 - x (- x) + 1 (2 - x))$

Step 3.3.1.3

Apply the distributive property.

$4 x^{2} = 50.5 (- x \cdot 2 - x (- x) + 1 \cdot 2 + 1 (- x))$

Step 3.3.2

Simplify and combine like terms.

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Step 3.3.2.1

Simplify each term.

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Step 3.3.2.1.1

Multiply $2$ by $- 1$ .

$4 x^{2} = 50.5 (- 2 x - x (- x) + 1 \cdot 2 + 1 (- x))$

Step 3.3.2.1.2

Rewrite using the commutative property of multiplication.

$4 x^{2} = 50.5 (- 2 x - 1 \cdot - 1 x \cdot x + 1 \cdot 2 + 1 (- x))$

Step 3.3.2.1.3

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

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Step 3.3.2.1.3.1

Move $x$ .

$4 x^{2} = 50.5 (- 2 x - 1 \cdot - 1 (x \cdot x) + 1 \cdot 2 + 1 (- x))$

Step 3.3.2.1.3.2

Multiply $x$ by $x$ .

$4 x^{2} = 50.5 (- 2 x - 1 \cdot - 1 x^{2} + 1 \cdot 2 + 1 (- x))$

Step 3.3.2.1.4

Multiply $- 1$ by $- 1$ .

$4 x^{2} = 50.5 (- 2 x + 1 x^{2} + 1 \cdot 2 + 1 (- x))$

Step 3.3.2.1.5

Multiply $x^{2}$ by $1$ .

$4 x^{2} = 50.5 (- 2 x + x^{2} + 1 \cdot 2 + 1 (- x))$

Step 3.3.2.1.6

Multiply $2$ by $1$ .

$4 x^{2} = 50.5 (- 2 x + x^{2} + 2 + 1 (- x))$

Step 3.3.2.1.7

Multiply $- x$ by $1$ .

$4 x^{2} = 50.5 (- 2 x + x^{2} + 2 - x)$

Step 3.3.2.2

Subtract $x$ from $- 2 x$ .

$4 x^{2} = 50.5 (- 3 x + x^{2} + 2)$

Step 3.3.3

Apply the distributive property.

$4 x^{2} = 50.5 (- 3 x) + 50.5 x^{2} + 50.5 \cdot 2$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Multiply $- 3$ by $50.5$ .

$4 x^{2} = - 151.5 x + 50.5 x^{2} + 50.5 \cdot 2$

Step 3.3.4.2

Multiply $50.5$ by $2$ .

$4 x^{2} = - 151.5 x + 50.5 x^{2} + 101$

Step 4

Solve the equation.

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Step 4.1

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

$- 151.5 x + 50.5 x^{2} + 101 = 4 x^{2}$

Step 4.2

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

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Step 4.2.1

Subtract $4 x^{2}$ from both sides of the equation.

$- 151.5 x + 50.5 x^{2} + 101 - 4 x^{2} = 0$

Step 4.2.2

Subtract $4 x^{2}$ from $50.5 x^{2}$ .

$- 151.5 x + 46.5 x^{2} + 101 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $0.5$ out of $- 151.5 x + 46.5 x^{2} + 101$ .

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Step 4.3.1.1

Factor $0.5$ out of $- 151.5 x$ .

$0.5 (- 303 x) + 46.5 x^{2} + 101 = 0$

Step 4.3.1.2

Factor $0.5$ out of $46.5 x^{2}$ .

$0.5 (- 303 x) + 0.5 (93 x^{2}) + 101 = 0$

Step 4.3.1.3

Factor $0.5$ out of $101$ .

$0.5 (- 303 x) + 0.5 (93 x^{2}) + 0.5 (202) = 0$

Step 4.3.1.4

Factor $0.5$ out of $0.5 (- 303 x) + 0.5 (93 x^{2})$ .

$0.5 (- 303 x + 93 x^{2}) + 0.5 (202) = 0$

Step 4.3.1.5

Factor $0.5$ out of $0.5 (- 303 x + 93 x^{2}) + 0.5 (202)$ .

$0.5 (- 303 x + 93 x^{2} + 202) = 0$

Step 4.3.2

Reorder terms.

$0.5 (93 x^{2} - 303 x + 202) = 0$

Step 4.4

Divide each term in $0.5 (93 x^{2} - 303 x + 202) = 0$ by $0.5$ and simplify.

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Step 4.4.1

Divide each term in $0.5 (93 x^{2} - 303 x + 202) = 0$ by $0.5$ .

$\frac{0.5 (93 x^{2} - 303 x + 202)}{0.5} = \frac{0}{0.5}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $0.5$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{0.5 (93 x^{2} - 303 x + 202)}{0.5} = \frac{0}{0.5}$

Step 4.4.2.1.2

Divide $93 x^{2} - 303 x + 202$ by $1$ .

$93 x^{2} - 303 x + 202 = \frac{0}{0.5}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Divide $0$ by $0.5$ .

$93 x^{2} - 303 x + 202 = 0$

Step 4.5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.6

Substitute the values $a = 93$ , $b = - 303$ , and $c = 202$ into the quadratic formula and solve for $x$ .

$\frac{303 \pm \sqrt{{(- 303)}^{2} - 4 \cdot (93 \cdot 202)}}{2 \cdot 93}$

Step 4.7

Simplify.

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Step 4.7.1

Simplify the numerator.

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Step 4.7.1.1

Raise $- 303$ to the power of $2$ .

$x = \frac{303 \pm \sqrt{91809 - 4 \cdot 93 \cdot 202}}{2 \cdot 93}$

Step 4.7.1.2

Multiply $- 4 \cdot 93 \cdot 202$ .

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Step 4.7.1.2.1

Multiply $- 4$ by $93$ .

$x = \frac{303 \pm \sqrt{91809 - 372 \cdot 202}}{2 \cdot 93}$

Step 4.7.1.2.2

Multiply $- 372$ by $202$ .

$x = \frac{303 \pm \sqrt{91809 - 75144}}{2 \cdot 93}$

Step 4.7.1.3

Subtract $75144$ from $91809$ .

$x = \frac{303 \pm \sqrt{16665}}{2 \cdot 93}$

Step 4.7.2

Multiply $2$ by $93$ .

$x = \frac{303 \pm \sqrt{16665}}{186}$

Step 4.8

The final answer is the combination of both solutions.

$x = \frac{303 + \sqrt{16665}}{186}, \frac{303 - \sqrt{16665}}{186}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{303 + \sqrt{16665}}{186}, \frac{303 - \sqrt{16665}}{186}$

Decimal Form:

$x = 2.32308059 \dots, 0.93498392 \dots$