Solve ДРУШТВО ЗА ГРАДЕЖ 1/(x - 1) - 1/x = 1/6 | 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

$$\frac { 1 } { x - 1 } - \frac { 1 } { x } = \frac { 1 } { 6 }$$

Solve for x

$x=-2$
$x=3$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to any of the values $0,1$ since division by zero is not defined. Multiply both sides of the equation by $6x\left(x-1\right)$, the least common multiple of $x-1,x,6$.

$$6x-\left(6x-6\right)=x\left(x-1\right)$$

To find the opposite of $6x-6$, find the opposite of each term.

$$6x-6x+6=x\left(x-1\right)$$

Combine $6x$ and $-6x$ to get $0$.

$$6=x\left(x-1\right)$$

Use the distributive property to multiply $x$ by $x-1$.

$$6=x^{2}-x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=6$$

Subtract $6$ from both sides.

$$x^{2}-x-6=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $1$ for $a$, $-1$ for $b$, and $-6$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-6\right)}}{2}$$

Multiply $-4$ times $-6$.

$$x=\frac{-\left(-1\right)±\sqrt{1+24}}{2}$$

Add $1$ to $24$.

$$x=\frac{-\left(-1\right)±\sqrt{25}}{2}$$

Take the square root of $25$.

$$x=\frac{-\left(-1\right)±5}{2}$$

The opposite of $-1$ is $1$.

$$x=\frac{1±5}{2}$$

Now solve the equation $x=\frac{1±5}{2}$ when $±$ is plus. Add $1$ to $5$.

$$x=\frac{6}{2}$$

Divide $6$ by $2$.

$$x=3$$

Now solve the equation $x=\frac{1±5}{2}$ when $±$ is minus. Subtract $5$ from $1$.

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

Divide $-4$ by $2$.

$$x=-2$$

The equation is now solved.

$$x=3$$ $$x=-2$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $0,1$ since division by zero is not defined. Multiply both sides of the equation by $6x\left(x-1\right)$, the least common multiple of $x-1,x,6$.

$$6x-\left(6x-6\right)=x\left(x-1\right)$$

To find the opposite of $6x-6$, find the opposite of each term.

$$6x-6x+6=x\left(x-1\right)$$

Combine $6x$ and $-6x$ to get $0$.

$$6=x\left(x-1\right)$$

Use the distributive property to multiply $x$ by $x-1$.

$$6=x^{2}-x$$

Swap sides so that all variable terms are on the left hand side.

$$x^{2}-x=6$$

Divide $-1$, the coefficient of the $x$ term, by $2$ to get $-\frac{1}{2}$. Then add the square of $-\frac{1}{2}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square $-\frac{1}{2}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=6+\frac{1}{4}$$

Add $6$ to $\frac{1}{4}$.

$$x^{2}-x+\frac{1}{4}=\frac{25}{4}$$

Factor $x^{2}-x+\frac{1}{4}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{5}{2}$$ $$x-\frac{1}{2}=-\frac{5}{2}$$

Add $\frac{1}{2}$ to both sides of the equation.

$$x=3$$ $$x=-2$$