Calculate: /(1) x+/(1) x-4 = /(10) 21

Expression: $$\frac { 1 } { x } + \frac { 1 } { x - 4 } = \frac { 10 } { 21 }$$

Variable $x$ cannot be equal to any of the values $0,4$ since division by zero is not defined. Multiply both sides of the equation by $21x\left(x-4\right)$, the least common multiple of $x,x-4,21$.


Combine $21x$ and $21x$ to get $42x$.


Use the distributive property to multiply $10x$ by $x-4$.


Subtract $10x^{2}$ from both sides.


Add $40x$ to both sides.


Combine $42x$ and $40x$ to get $82x$.


All equations of the form $ax^{2}+bx+c=0$ can be solved using the quadratic formula: $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$. The quadratic formula gives two solutions, one when $±$ is addition and one when it is subtraction.


This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-10$ for $a$, $82$ for $b$, and $-84$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.


Square $82$.


Multiply $-4$ times $-10$.


Multiply $40$ times $-84$.


Add $6724$ to $-3360$.


Take the square root of $3364$.


Multiply $2$ times $-10$.


Now solve the equation $x=\frac{-82±58}{-20}$ when $±$ is plus. Add $-82$ to $58$.


Reduce the fraction $\frac{-24}{-20}$ to lowest terms by extracting and canceling out $4$.


Now solve the equation $x=\frac{-82±58}{-20}$ when $±$ is minus. Subtract $58$ from $-82$.


Divide $-140$ by $-20$.


The equation is now solved.

$$x=\frac{6}{5}$$ $$x=7$$