Solve Century : 1/((x - 3)(x + 2)) + 3/((x + 2)(4 - x)) * 2/((x - 3)(x - 4)) | Equatix - Math Solver

Fractions & Decimals

Ratios & Proportions

Basic Number Theory

Integers & Absolute Values

Factors & Multiples

Prime Factorization

Basic Equations & Inequalities

Coordinate Plane & Graphing

Linear Equations & Inequalities

Quadratic Equations

Polynomials & Factoring

Functions & Graphs

Systems of Equations

Trigonometric Functions

Trigonometric Identities & Equations

Right Triangle Trigonometry

Graphing Trigonometric Functions

Limits & Continuity

Derivatives & Applications

Integrals & Applications

Series & Sequences

Multivariable Calculus

Functions & Graphs

Polynomial & Rational Functions

Exponential & Logarithmic Functions

Sequences & Series

Descriptive Statistics

Normal Distribution

Hypothesis Testing

Regression Analysis

Linear Programming

Sets & Counting

Eigenvalues & Eigenvectors

Linear Transformations

Atomic Structure

Thermochemistry

Example

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

a⋅(b-2)=3b

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

a⋅(b-2)=3b

Question

$$\frac { 1 } { ( x - 3 ) ( x + 2 ) } + \frac { 3 } { ( x + 1 ) ( 4 - x ) } + \frac { 2 } { ( x - 3 ) ( x - 2 ) }$$

Evaluate

$-\frac{2\left(x^{2}-3x-22\right)}{\left(4-x\right)\left(x-3\right)\left(x+1\right)\left(x^{2}-4\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x+2\right)$ and $\left(x+1\right)\left(4-x\right)$ is $\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$. Multiply $\frac{1}{\left(x-3\right)\left(x+2\right)}$ times $\frac{\left(x+1\right)\left(-x+4\right)}{\left(x+1\right)\left(-x+4\right)}$. Multiply $\frac{3}{\left(x+1\right)\left(4-x\right)}$ times $\frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+2\right)}$.

$$\frac{\left(x+1\right)\left(-x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{3\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

Since $\frac{\left(x+1\right)\left(-x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ and $\frac{3\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)\left(-x+4\right)+3\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

Do the multiplications in $\left(x+1\right)\left(-x+4\right)+3\left(x-3\right)\left(x+2\right)$.

$$\frac{-x^{2}+4x+4-x+3x^{2}+6x-9x-18}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

Combine like terms in $-x^{2}+4x+4-x+3x^{2}+6x-9x-18$.

$$\frac{2x^{2}-14}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$ and $\left(x-3\right)\left(x-2\right)$ is $\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$. Multiply $\frac{2x^{2}-14}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ times $\frac{x-2}{x-2}$. Multiply $\frac{2}{\left(x-3\right)\left(x-2\right)}$ times $\frac{\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$.

$$\frac{\left(2x^{2}-14\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Since $\frac{\left(2x^{2}-14\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ and $\frac{2\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{2}-14\right)\left(x-2\right)+2\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Do the multiplications in $\left(2x^{2}-14\right)\left(x-2\right)+2\left(x+1\right)\left(x+2\right)\left(-x+4\right)$.

$$\frac{2x^{3}-4x^{2}-14x+28-2x^{3}+4x^{2}+16x-2x^{2}+4x+16}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Combine like terms in $2x^{3}-4x^{2}-14x+28-2x^{3}+4x^{2}+16x-2x^{2}+4x+16$.

$$\frac{-2x^{2}+6x+44}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Expand $\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$.

$$\frac{-2x^{2}+6x+44}{-x^{5}+6x^{4}-x^{3}-36x^{2}+20x+48}$$

Expand

$-\frac{2\left(x^{2}-3x-22\right)}{\left(4-x\right)\left(x-3\right)\left(x+1\right)\left(x^{2}-4\right)}$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x+2\right)$ and $\left(x+1\right)\left(4-x\right)$ is $\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$. Multiply $\frac{1}{\left(x-3\right)\left(x+2\right)}$ times $\frac{\left(x+1\right)\left(-x+4\right)}{\left(x+1\right)\left(-x+4\right)}$. Multiply $\frac{3}{\left(x+1\right)\left(4-x\right)}$ times $\frac{\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+2\right)}$.

$$\frac{\left(x+1\right)\left(-x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{3\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

Since $\frac{\left(x+1\right)\left(-x+4\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ and $\frac{3\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(x+1\right)\left(-x+4\right)+3\left(x-3\right)\left(x+2\right)}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

Do the multiplications in $\left(x+1\right)\left(-x+4\right)+3\left(x-3\right)\left(x+2\right)$.

$$\frac{-x^{2}+4x+4-x+3x^{2}+6x-9x-18}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

Combine like terms in $-x^{2}+4x+4-x+3x^{2}+6x-9x-18$.

$$\frac{2x^{2}-14}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2}{\left(x-3\right)\left(x-2\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$ and $\left(x-3\right)\left(x-2\right)$ is $\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$. Multiply $\frac{2x^{2}-14}{\left(x-3\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ times $\frac{x-2}{x-2}$. Multiply $\frac{2}{\left(x-3\right)\left(x-2\right)}$ times $\frac{\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$.

$$\frac{\left(2x^{2}-14\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}+\frac{2\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Since $\frac{\left(2x^{2}-14\right)\left(x-2\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ and $\frac{2\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$ have the same denominator, add them by adding their numerators.

$$\frac{\left(2x^{2}-14\right)\left(x-2\right)+2\left(x+1\right)\left(x+2\right)\left(-x+4\right)}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Do the multiplications in $\left(2x^{2}-14\right)\left(x-2\right)+2\left(x+1\right)\left(x+2\right)\left(-x+4\right)$.

$$\frac{2x^{3}-4x^{2}-14x+28-2x^{3}+4x^{2}+16x-2x^{2}+4x+16}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Combine like terms in $2x^{3}-4x^{2}-14x+28-2x^{3}+4x^{2}+16x-2x^{2}+4x+16$.

$$\frac{-2x^{2}+6x+44}{\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)}$$

Expand $\left(x-3\right)\left(x-2\right)\left(x+1\right)\left(x+2\right)\left(-x+4\right)$.

$$\frac{-2x^{2}+6x+44}{-x^{5}+6x^{4}-x^{3}-36x^{2}+20x+48}$$