How to solve 4(x)/(2x-4)

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

Solve 4(x)/(2x-4) | Equatix

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Question

$$4 \frac{ x }{ 2x-4 }$$

Evaluate

$\frac{2x}{x-2}$

Short Solution Steps

Express $4\times \frac{x}{2x-4}$ as a single fraction.

$$\frac{4x}{2x-4}$$

Factor the expressions that are not already factored.

$$\frac{4x}{2\left(x-2\right)}$$

Cancel out $2$ in both numerator and denominator.

$$\frac{2x}{x-2}$$

Show Solution

Differentiate w.r.t. x

$-\frac{4}{\left(x-2\right)^{2}}$

Steps Using Derivative Rule for Quotient

Express $4\times \frac{x}{2x-4}$ as a single fraction.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x}{2x-4})$$

Factor the expressions that are not already factored in $\frac{4x}{2x-4}$.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{4x}{2\left(x-2\right)})$$

Cancel out $2$ in both numerator and denominator.

$$\frac{\mathrm{d}}{\mathrm{d}x}(\frac{2x}{x-2})$$

For any two differentiable functions, the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the denominator squared.

$$\frac{\left(x^{1}-2\right)\frac{\mathrm{d}}{\mathrm{d}x}(2x^{1})-2x^{1}\frac{\mathrm{d}}{\mathrm{d}x}(x^{1}-2)}{\left(x^{1}-2\right)^{2}}$$

The derivative of a polynomial is the sum of the derivatives of its terms. The derivative of a constant term is $0$. The derivative of $ax^{n}$ is $nax^{n-1}$.

$$\frac{\left(x^{1}-2\right)\times 2x^{1-1}-2x^{1}x^{1-1}}{\left(x^{1}-2\right)^{2}}$$

Do the arithmetic.

$$\frac{\left(x^{1}-2\right)\times 2x^{0}-2x^{1}x^{0}}{\left(x^{1}-2\right)^{2}}$$

Expand using distributive property.

$$\frac{x^{1}\times 2x^{0}-2\times 2x^{0}-2x^{1}x^{0}}{\left(x^{1}-2\right)^{2}}$$

To multiply powers of the same base, add their exponents.

$$\frac{2x^{1}-2\times 2x^{0}-2x^{1}}{\left(x^{1}-2\right)^{2}}$$

Do the arithmetic.

$$\frac{2x^{1}-4x^{0}-2x^{1}}{\left(x^{1}-2\right)^{2}}$$

Combine like terms.

$$\frac{\left(2-2\right)x^{1}-4x^{0}}{\left(x^{1}-2\right)^{2}}$$

Subtract $2$ from $2$.

$$\frac{-4x^{0}}{\left(x^{1}-2\right)^{2}}$$

For any term $t$, $t^{1}=t$.

$$\frac{-4x^{0}}{\left(x-2\right)^{2}}$$

For any term $t$ except $0$, $t^{0}=1$.

$$\frac{-4}{\left(x-2\right)^{2}}$$