Solve by Factoring (4x)/(x+5)+20/x=100/(x^2+5x)

$\frac{4 x}{x + 5} + \frac{20}{x} = \frac{100}{x^{2} + 5 x}$

Step 1

Subtract $\frac{100}{x^{2} + 5 x}$ from both sides of the equation.

$\frac{4 x}{x + 5} + \frac{20}{x} - \frac{100}{x^{2} + 5 x} = 0$

Step 2

Simplify $\frac{4 x}{x + 5} + \frac{20}{x} - \frac{100}{x^{2} + 5 x}$ .

Factor $x$ out of $x^{2} + 5 x$ .

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

$\frac{4 x}{x + 5} + \frac{20}{x} - \frac{100}{x \cdot x + 5 x} = 0$

Factor $x$ out of $5 x$ .

$\frac{4 x}{x + 5} + \frac{20}{x} - \frac{100}{x \cdot x + x \cdot 5} = 0$

Factor $x$ out of $x \cdot x + x \cdot 5$ .

$\frac{4 x}{x + 5} + \frac{20}{x} - \frac{100}{x (x + 5)} = 0$

To write $\frac{4 x}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 x}{x + 5} \cdot \frac{x}{x} + \frac{20}{x} - \frac{100}{x (x + 5)} = 0$

To write $\frac{20}{x}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

$\frac{4 x}{x + 5} \cdot \frac{x}{x} + \frac{20}{x} \cdot \frac{x + 5}{x + 5} - \frac{100}{x (x + 5)} = 0$

Write each expression with a common denominator of $(x + 5) x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 x}{x + 5}$ by $\frac{x}{x}$ .

$\frac{4 x \cdot x}{(x + 5) x} + \frac{20}{x} \cdot \frac{x + 5}{x + 5} - \frac{100}{x (x + 5)} = 0$

Multiply $\frac{20}{x}$ by $\frac{x + 5}{x + 5}$ .

$\frac{4 x \cdot x}{(x + 5) x} + \frac{20 (x + 5)}{x (x + 5)} - \frac{100}{x (x + 5)} = 0$

Reorder the factors of $(x + 5) x$ .

$\frac{4 x \cdot x}{x (x + 5)} + \frac{20 (x + 5)}{x (x + 5)} - \frac{100}{x (x + 5)} = 0$

Combine the numerators over the common denominator.

$\frac{4 x \cdot x + 20 (x + 5)}{x (x + 5)} - \frac{100}{x (x + 5)} = 0$

Combine the numerators over the common denominator.

$\frac{4 x \cdot x + 20 (x + 5) - 100}{x (x + 5)} = 0$

Simplify each term.

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

Move $x$ .

$\frac{4 (x \cdot x) + 20 (x + 5) - 100}{x (x + 5)} = 0$

Multiply $x$ by $x$ .

$\frac{4 x^{2} + 20 (x + 5) - 100}{x (x + 5)} = 0$

Apply the distributive property.

$\frac{4 x^{2} + 20 x + 20 \cdot 5 - 100}{x (x + 5)} = 0$

Multiply $20$ by $5$ .

$\frac{4 x^{2} + 20 x + 100 - 100}{x (x + 5)} = 0$

Combine the opposite terms in $4 x^{2} + 20 x + 100 - 100$ .

Subtract $100$ from $100$ .

$\frac{4 x^{2} + 20 x + 0}{x (x + 5)} = 0$

Add $4 x^{2} + 20 x$ and $0$ .

$\frac{4 x^{2} + 20 x}{x (x + 5)} = 0$

Factor $4 x$ out of $4 x^{2} + 20 x$ .

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

$\frac{4 x (x) + 20 x}{x (x + 5)} = 0$

Factor $4 x$ out of $20 x$ .

$\frac{4 x (x) + 4 x (5)}{x (x + 5)} = 0$

Factor $4 x$ out of $4 x (x) + 4 x (5)$ .

$\frac{4 x (x + 5)}{x (x + 5)} = 0$

Cancel the common factor of $x$ .

Cancel the common factor.

$\frac{4 x (x + 5)}{x (x + 5)} = 0$

Rewrite the expression.

$\frac{4 (x + 5)}{x + 5} = 0$

Cancel the common factor of $x + 5$ .

Cancel the common factor.

$\frac{4 (x + 5)}{x + 5} = 0$

Divide $4$ by $1$ .

$4 = 0$

Step 3

Since $4 \neq 0$ , there are no solutions.

No solution

Solve by Factoring (4x)/(x+5)+20/x=100/(x^2+5x)

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