Solve Find the solution set of (x - 4)/(x + 4) + (x + 4)/(x - 4) = 4/5 | 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

$$\frac { x - 4 } { x + 4 } + \frac { x + 4 } { x - 4 } = \frac { 4 } { 5 }$$

Solve for x (complex solution)

$x=-\frac{4\sqrt{21}i}{3}\approx -0-6.110100927i$
$x=\frac{4\sqrt{21}i}{3}\approx 6.110100927i$

Steps by Finding Square Root

Steps Using the Quadratic Formula

Steps by Finding Square Root

Variable $x$ cannot be equal to any of the values $-4,4$ since division by zero is not defined. Multiply both sides of the equation by $5\left(x-4\right)\left(x+4\right)$, the least common multiple of $x+4,x-4,5$.

$$\left(5x-20\right)\left(x-4\right)+\left(5x+20\right)\left(x+4\right)=4\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply $5x-20$ by $x-4$ and combine like terms.

$$5x^{2}-40x+80+\left(5x+20\right)\left(x+4\right)=4\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply $5x+20$ by $x+4$ and combine like terms.

$$5x^{2}-40x+80+5x^{2}+40x+80=4\left(x-4\right)\left(x+4\right)$$

Combine $5x^{2}$ and $5x^{2}$ to get $10x^{2}$.

$$10x^{2}-40x+80+40x+80=4\left(x-4\right)\left(x+4\right)$$

Combine $-40x$ and $40x$ to get $0$.

$$10x^{2}+80+80=4\left(x-4\right)\left(x+4\right)$$

Add $80$ and $80$ to get $160$.

$$10x^{2}+160=4\left(x-4\right)\left(x+4\right)$$

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

$$10x^{2}+160=\left(4x-16\right)\left(x+4\right)$$

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

$$10x^{2}+160=4x^{2}-64$$

Subtract $4x^{2}$ from both sides.

$$10x^{2}+160-4x^{2}=-64$$

Combine $10x^{2}$ and $-4x^{2}$ to get $6x^{2}$.

$$6x^{2}+160=-64$$

Subtract $160$ from both sides.

$$6x^{2}=-64-160$$

Subtract $160$ from $-64$ to get $-224$.

$$6x^{2}=-224$$

Divide both sides by $6$.

$$x^{2}=\frac{-224}{6}$$

Reduce the fraction $\frac{-224}{6}$ to lowest terms by extracting and canceling out $2$.

$$x^{2}=-\frac{112}{3}$$

The equation is now solved.

$$x=\frac{4\sqrt{21}i}{3}$$ $$x=-\frac{4\sqrt{21}i}{3}$$

Steps Using the Quadratic Formula

Variable $x$ cannot be equal to any of the values $-4,4$ since division by zero is not defined. Multiply both sides of the equation by $5\left(x-4\right)\left(x+4\right)$, the least common multiple of $x+4,x-4,5$.

$$\left(5x-20\right)\left(x-4\right)+\left(5x+20\right)\left(x+4\right)=4\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply $5x-20$ by $x-4$ and combine like terms.

$$5x^{2}-40x+80+\left(5x+20\right)\left(x+4\right)=4\left(x-4\right)\left(x+4\right)$$

Use the distributive property to multiply $5x+20$ by $x+4$ and combine like terms.

$$5x^{2}-40x+80+5x^{2}+40x+80=4\left(x-4\right)\left(x+4\right)$$

Combine $5x^{2}$ and $5x^{2}$ to get $10x^{2}$.

$$10x^{2}-40x+80+40x+80=4\left(x-4\right)\left(x+4\right)$$

Combine $-40x$ and $40x$ to get $0$.

$$10x^{2}+80+80=4\left(x-4\right)\left(x+4\right)$$

Add $80$ and $80$ to get $160$.

$$10x^{2}+160=4\left(x-4\right)\left(x+4\right)$$

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

$$10x^{2}+160=\left(4x-16\right)\left(x+4\right)$$

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

$$10x^{2}+160=4x^{2}-64$$

Subtract $4x^{2}$ from both sides.

$$10x^{2}+160-4x^{2}=-64$$

Combine $10x^{2}$ and $-4x^{2}$ to get $6x^{2}$.

$$6x^{2}+160=-64$$

Add $64$ to both sides.

$$6x^{2}+160+64=0$$

Add $160$ and $64$ to get $224$.

$$6x^{2}+224=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $6$ for $a$, $0$ for $b$, and $224$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{0±\sqrt{0^{2}-4\times 6\times 224}}{2\times 6}$$

Square $0$.

$$x=\frac{0±\sqrt{-4\times 6\times 224}}{2\times 6}$$

Multiply $-4$ times $6$.

$$x=\frac{0±\sqrt{-24\times 224}}{2\times 6}$$

Multiply $-24$ times $224$.

$$x=\frac{0±\sqrt{-5376}}{2\times 6}$$

Take the square root of $-5376$.

$$x=\frac{0±16\sqrt{21}i}{2\times 6}$$

Multiply $2$ times $6$.

$$x=\frac{0±16\sqrt{21}i}{12}$$

Now solve the equation $x=\frac{0±16\sqrt{21}i}{12}$ when $±$ is plus.

$$x=\frac{4\sqrt{21}i}{3}$$

Now solve the equation $x=\frac{0±16\sqrt{21}i}{12}$ when $±$ is minus.

$$x=-\frac{4\sqrt{21}i}{3}$$

The equation is now solved.

$$x=\frac{4\sqrt{21}i}{3}$$ $$x=-\frac{4\sqrt{21}i}{3}$$