Solve (x + 8)/(x - 8) - 24/(x - 4) = 2 | 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

$$\frac{x+8}{x-8}-\frac{24}{x-4}=2$$

Solve for x

$x=-8$
$x=12$

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 $4,8$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-8\right)\left(x-4\right)$, the least common multiple of $x-8,x-4$.

$$\left(x-4\right)\left(x+8\right)-\left(x-8\right)\times 24=2\left(x-8\right)\left(x-4\right)$$

Use the distributive property to multiply $x-4$ by $x+8$ and combine like terms.

$$x^{2}+4x-32-\left(x-8\right)\times 24=2\left(x-8\right)\left(x-4\right)$$

Use the distributive property to multiply $x-8$ by $24$.

$$x^{2}+4x-32-\left(24x-192\right)=2\left(x-8\right)\left(x-4\right)$$

To find the opposite of $24x-192$, find the opposite of each term.

$$x^{2}+4x-32-24x+192=2\left(x-8\right)\left(x-4\right)$$

Combine $4x$ and $-24x$ to get $-20x$.

$$x^{2}-20x-32+192=2\left(x-8\right)\left(x-4\right)$$

Add $-32$ and $192$ to get $160$.

$$x^{2}-20x+160=2\left(x-8\right)\left(x-4\right)$$

Use the distributive property to multiply $2$ by $x-8$.

$$x^{2}-20x+160=\left(2x-16\right)\left(x-4\right)$$

Use the distributive property to multiply $2x-16$ by $x-4$ and combine like terms.

$$x^{2}-20x+160=2x^{2}-24x+64$$

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

$$x^{2}-20x+160-2x^{2}=-24x+64$$

Combine $x^{2}$ and $-2x^{2}$ to get $-x^{2}$.

$$-x^{2}-20x+160=-24x+64$$

Add $24x$ to both sides.

$$-x^{2}-20x+160+24x=64$$

Combine $-20x$ and $24x$ to get $4x$.

$$-x^{2}+4x+160=64$$

Subtract $64$ from both sides.

$$-x^{2}+4x+160-64=0$$

Subtract $64$ from $160$ to get $96$.

$$-x^{2}+4x+96=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1$ for $a$, $4$ for $b$, and $96$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 96}}{2\left(-1\right)}$$

Square $4$.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 96}}{2\left(-1\right)}$$

Multiply $-4$ times $-1$.

$$x=\frac{-4±\sqrt{16+4\times 96}}{2\left(-1\right)}$$

Multiply $4$ times $96$.

$$x=\frac{-4±\sqrt{16+384}}{2\left(-1\right)}$$

Add $16$ to $384$.

$$x=\frac{-4±\sqrt{400}}{2\left(-1\right)}$$

Take the square root of $400$.

$$x=\frac{-4±20}{2\left(-1\right)}$$

Multiply $2$ times $-1$.

$$x=\frac{-4±20}{-2}$$

Now solve the equation $x=\frac{-4±20}{-2}$ when $±$ is plus. Add $-4$ to $20$.

$$x=\frac{16}{-2}$$

Divide $16$ by $-2$.

$$x=-8$$

Now solve the equation $x=\frac{-4±20}{-2}$ when $±$ is minus. Subtract $20$ from $-4$.

$$x=-\frac{24}{-2}$$

Divide $-24$ by $-2$.

$$x=12$$

The equation is now solved.

$$x=-8$$ $$x=12$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $4,8$ since division by zero is not defined. Multiply both sides of the equation by $\left(x-8\right)\left(x-4\right)$, the least common multiple of $x-8,x-4$.

$$\left(x-4\right)\left(x+8\right)-\left(x-8\right)\times 24=2\left(x-8\right)\left(x-4\right)$$

Use the distributive property to multiply $x-4$ by $x+8$ and combine like terms.

$$x^{2}+4x-32-\left(x-8\right)\times 24=2\left(x-8\right)\left(x-4\right)$$

Use the distributive property to multiply $x-8$ by $24$.

$$x^{2}+4x-32-\left(24x-192\right)=2\left(x-8\right)\left(x-4\right)$$

To find the opposite of $24x-192$, find the opposite of each term.

$$x^{2}+4x-32-24x+192=2\left(x-8\right)\left(x-4\right)$$

Combine $4x$ and $-24x$ to get $-20x$.

$$x^{2}-20x-32+192=2\left(x-8\right)\left(x-4\right)$$

Add $-32$ and $192$ to get $160$.

$$x^{2}-20x+160=2\left(x-8\right)\left(x-4\right)$$

Use the distributive property to multiply $2$ by $x-8$.

$$x^{2}-20x+160=\left(2x-16\right)\left(x-4\right)$$

Use the distributive property to multiply $2x-16$ by $x-4$ and combine like terms.

$$x^{2}-20x+160=2x^{2}-24x+64$$

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

$$x^{2}-20x+160-2x^{2}=-24x+64$$

Combine $x^{2}$ and $-2x^{2}$ to get $-x^{2}$.

$$-x^{2}-20x+160=-24x+64$$

Add $24x$ to both sides.

$$-x^{2}-20x+160+24x=64$$

Combine $-20x$ and $24x$ to get $4x$.

$$-x^{2}+4x+160=64$$

Subtract $160$ from both sides.

$$-x^{2}+4x=64-160$$

Subtract $160$ from $64$ to get $-96$.

$$-x^{2}+4x=-96$$

Divide both sides by $-1$.

$$\frac{-x^{2}+4x}{-1}=-\frac{96}{-1}$$

Dividing by $-1$ undoes the multiplication by $-1$.

$$x^{2}+\frac{4}{-1}x=-\frac{96}{-1}$$

Divide $4$ by $-1$.

$$x^{2}-4x=-\frac{96}{-1}$$

Divide $-96$ by $-1$.

$$x^{2}-4x=96$$

Divide $-4$, the coefficient of the $x$ term, by $2$ to get $-2$. Then add the square of $-2$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=96+\left(-2\right)^{2}$$

Square $-2$.

$$x^{2}-4x+4=96+4$$

Add $96$ to $4$.

$$x^{2}-4x+4=100$$

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

$$\left(x-2\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-2=10$$ $$x-2=-10$$

Add $2$ to both sides of the equation.

$$x=12$$ $$x=-8$$