Solve 3/(x + 2) - 1/x = 1/16 * 4/15 | 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

$$\frac{3}{(x+2)}-\frac{1}{x}=\frac{1}{16}\frac{4}{15}$$

Solve for x

$x=\sqrt{3361}+59\approx 116.974132163$
$x=59-\sqrt{3361}\approx 1.025867837$

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 $-2,0$ since division by zero is not defined. Multiply both sides of the equation by $240x\left(x+2\right)$, the least common multiple of $x+2,x,16,15$.

$$240x\times 3-\left(240x+480\right)=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

Multiply $240$ and $3$ to get $720$.

$$720x-\left(240x+480\right)=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

To find the opposite of $240x+480$, find the opposite of each term.

$$720x-240x-480=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

Combine $720x$ and $-240x$ to get $480x$.

$$480x-480=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

Multiply $\frac{1}{16}$ and $\frac{4}{15}$ to get $\frac{1}{60}$.

$$480x-480=\frac{1}{60}\times 240x\left(x+2\right)$$

Multiply $\frac{1}{60}$ and $240$ to get $4$.

$$480x-480=4x\left(x+2\right)$$

Use the distributive property to multiply $4x$ by $x+2$.

$$480x-480=4x^{2}+8x$$

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

$$480x-480-4x^{2}=8x$$

Subtract $8x$ from both sides.

$$480x-480-4x^{2}-8x=0$$

Combine $480x$ and $-8x$ to get $472x$.

$$472x-480-4x^{2}=0$$

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.

$$-4x^{2}+472x-480=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-4$ for $a$, $472$ for $b$, and $-480$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-472±\sqrt{472^{2}-4\left(-4\right)\left(-480\right)}}{2\left(-4\right)}$$

Square $472$.

$$x=\frac{-472±\sqrt{222784-4\left(-4\right)\left(-480\right)}}{2\left(-4\right)}$$

Multiply $-4$ times $-4$.

$$x=\frac{-472±\sqrt{222784+16\left(-480\right)}}{2\left(-4\right)}$$

Multiply $16$ times $-480$.

$$x=\frac{-472±\sqrt{222784-7680}}{2\left(-4\right)}$$

Add $222784$ to $-7680$.

$$x=\frac{-472±\sqrt{215104}}{2\left(-4\right)}$$

Take the square root of $215104$.

$$x=\frac{-472±8\sqrt{3361}}{2\left(-4\right)}$$

Multiply $2$ times $-4$.

$$x=\frac{-472±8\sqrt{3361}}{-8}$$

Now solve the equation $x=\frac{-472±8\sqrt{3361}}{-8}$ when $±$ is plus. Add $-472$ to $8\sqrt{3361}$.

$$x=\frac{8\sqrt{3361}-472}{-8}$$

Divide $-472+8\sqrt{3361}$ by $-8$.

$$x=59-\sqrt{3361}$$

Now solve the equation $x=\frac{-472±8\sqrt{3361}}{-8}$ when $±$ is minus. Subtract $8\sqrt{3361}$ from $-472$.

$$x=\frac{-8\sqrt{3361}-472}{-8}$$

Divide $-472-8\sqrt{3361}$ by $-8$.

$$x=\sqrt{3361}+59$$

The equation is now solved.

$$x=59-\sqrt{3361}$$ $$x=\sqrt{3361}+59$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $-2,0$ since division by zero is not defined. Multiply both sides of the equation by $240x\left(x+2\right)$, the least common multiple of $x+2,x,16,15$.

$$240x\times 3-\left(240x+480\right)=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

Multiply $240$ and $3$ to get $720$.

$$720x-\left(240x+480\right)=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

To find the opposite of $240x+480$, find the opposite of each term.

$$720x-240x-480=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

Combine $720x$ and $-240x$ to get $480x$.

$$480x-480=\frac{1}{16}\times \frac{4}{15}\times 240x\left(x+2\right)$$

Multiply $\frac{1}{16}$ and $\frac{4}{15}$ to get $\frac{1}{60}$.

$$480x-480=\frac{1}{60}\times 240x\left(x+2\right)$$

Multiply $\frac{1}{60}$ and $240$ to get $4$.

$$480x-480=4x\left(x+2\right)$$

Use the distributive property to multiply $4x$ by $x+2$.

$$480x-480=4x^{2}+8x$$

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

$$480x-480-4x^{2}=8x$$

Subtract $8x$ from both sides.

$$480x-480-4x^{2}-8x=0$$

Combine $480x$ and $-8x$ to get $472x$.

$$472x-480-4x^{2}=0$$

Add $480$ to both sides. Anything plus zero gives itself.

$$472x-4x^{2}=480$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form $x^{2}+bx=c$.

$$-4x^{2}+472x=480$$

Divide both sides by $-4$.

$$\frac{-4x^{2}+472x}{-4}=\frac{480}{-4}$$

Dividing by $-4$ undoes the multiplication by $-4$.

$$x^{2}+\frac{472}{-4}x=\frac{480}{-4}$$

Divide $472$ by $-4$.

$$x^{2}-118x=\frac{480}{-4}$$

Divide $480$ by $-4$.

$$x^{2}-118x=-120$$

Divide $-118$, the coefficient of the $x$ term, by $2$ to get $-59$. Then add the square of $-59$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-118x+\left(-59\right)^{2}=-120+\left(-59\right)^{2}$$

Square $-59$.

$$x^{2}-118x+3481=-120+3481$$

Add $-120$ to $3481$.

$$x^{2}-118x+3481=3361$$

Factor $x^{2}-118x+3481$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x-59\right)^{2}=3361$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-59\right)^{2}}=\sqrt{3361}$$

Simplify.

$$x-59=\sqrt{3361}$$ $$x-59=-\sqrt{3361}$$

Add $59$ to both sides of the equation.

$$x=\sqrt{3361}+59$$ $$x=59-\sqrt{3361}$$