Solve (a ^ 2 - x ^ 2)/(b ^ 2 - 16) * (b + 4)/(a - x) + x/(4 - b) | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$\frac{a^{2}-x^{2}}{b^{2}-16}\cdot\frac{b+4}{a-x}+\frac{x}{4-b}$$

Evaluate

$\frac{a}{b-4}$

Short Solution Steps

Multiply $\frac{a^{2}-x^{2}}{b^{2}-16}$ times $\frac{b+4}{a-x}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b^{2}-16\right)\left(a-x\right)}+\frac{x}{4-b}$$

Factor $\left(b^{2}-16\right)\left(a-x\right)$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(b-4\right)\left(b+4\right)\left(-x+a\right)$ and $4-b$ is $\left(b-4\right)\left(-b-4\right)\left(-x+a\right)$. Multiply $\frac{\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b-4\right)\left(b+4\right)\left(-x+a\right)}$ times $\frac{-1}{-1}$. Multiply $\frac{x}{4-b}$ times $\frac{-\left(-b-4\right)\left(-x+a\right)}{-\left(-b-4\right)\left(-x+a\right)}$.

$$\frac{-\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}+\frac{x\left(-1\right)\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Since $\frac{-\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$ and $\frac{x\left(-1\right)\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-\left(a^{2}-x^{2}\right)\left(b+4\right)+x\left(-1\right)\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Do the multiplications in $-\left(a^{2}-x^{2}\right)\left(b+4\right)+x\left(-1\right)\left(-b-4\right)\left(-x+a\right)$.

$$\frac{-a^{2}b-4a^{2}+x^{2}b+4x^{2}-x^{2}b+xba-4x^{2}+4xa}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Combine like terms in $-a^{2}b-4a^{2}+x^{2}b+4x^{2}-x^{2}b+xba-4x^{2}+4xa$.

$$\frac{-a^{2}b+xba-4a^{2}+4xa}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Factor the expressions that are not already factored in $\frac{-a^{2}b+xba-4a^{2}+4xa}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$.

$$\frac{a\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Cancel out $\left(-b-4\right)\left(-x+a\right)$ in both numerator and denominator.

$$\frac{a}{b-4}$$

Expand

$-\frac{a}{4-b}$

Short Solution Steps

Multiply $\frac{a^{2}-x^{2}}{b^{2}-16}$ times $\frac{b+4}{a-x}$ by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b^{2}-16\right)\left(a-x\right)}+\frac{x}{4-b}$$

Factor $\left(b^{2}-16\right)\left(a-x\right)$.

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(b-4\right)\left(b+4\right)\left(-x+a\right)$ and $4-b$ is $\left(b-4\right)\left(-b-4\right)\left(-x+a\right)$. Multiply $\frac{\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b-4\right)\left(b+4\right)\left(-x+a\right)}$ times $\frac{-1}{-1}$. Multiply $\frac{x}{4-b}$ times $\frac{-\left(-b-4\right)\left(-x+a\right)}{-\left(-b-4\right)\left(-x+a\right)}$.

$$\frac{-\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}+\frac{x\left(-1\right)\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Since $\frac{-\left(a^{2}-x^{2}\right)\left(b+4\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$ and $\frac{x\left(-1\right)\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{-\left(a^{2}-x^{2}\right)\left(b+4\right)+x\left(-1\right)\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Do the multiplications in $-\left(a^{2}-x^{2}\right)\left(b+4\right)+x\left(-1\right)\left(-b-4\right)\left(-x+a\right)$.

$$\frac{-a^{2}b-4a^{2}+x^{2}b+4x^{2}-x^{2}b+xba-4x^{2}+4xa}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Combine like terms in $-a^{2}b-4a^{2}+x^{2}b+4x^{2}-x^{2}b+xba-4x^{2}+4xa$.

$$\frac{-a^{2}b+xba-4a^{2}+4xa}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Factor the expressions that are not already factored in $\frac{-a^{2}b+xba-4a^{2}+4xa}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$.

$$\frac{a\left(-b-4\right)\left(-x+a\right)}{\left(b-4\right)\left(-b-4\right)\left(-x+a\right)}$$

Cancel out $\left(-b-4\right)\left(-x+a\right)$ in both numerator and denominator.

$$\frac{a}{b-4}$$