# 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 $x,x-4,21$.

$$21x-84+21x=10x\left(x-4\right)$$

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

$$42x-84=10x\left(x-4\right)$$

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

$$42x-84=10x^{2}-40x$$

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

$$42x-84-10x^{2}=-40x$$

Add $40x$ to both sides.

$$42x-84-10x^{2}+40x=0$$

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

$$82x-84-10x^{2}=0$$

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.

$$-10x^{2}+82x-84=0$$

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, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-82±\sqrt{82^{2}-4\left(-10\right)\left(-84\right)}}{2\left(-10\right)}$$

Square $82$.

$$x=\frac{-82±\sqrt{6724-4\left(-10\right)\left(-84\right)}}{2\left(-10\right)}$$

Multiply $-4$ times $-10$.

$$x=\frac{-82±\sqrt{6724+40\left(-84\right)}}{2\left(-10\right)}$$

Multiply $40$ times $-84$.

$$x=\frac{-82±\sqrt{6724-3360}}{2\left(-10\right)}$$

Add $6724$ to $-3360$.

$$x=\frac{-82±\sqrt{3364}}{2\left(-10\right)}$$

Take the square root of $3364$.

$$x=\frac{-82±58}{2\left(-10\right)}$$

Multiply $2$ times $-10$.

$$x=\frac{-82±58}{-20}$$

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

$$x=\frac{-24}{-20}$$

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

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

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

$$x=\frac{-140}{-20}$$

Divide $-140$ by $-20$.

$$x=7$$

The equation is now solved.

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

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