Solve 1/(7 - x) - 1/(4 + x) = 11/30 | 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

$$\frac{1}{7-x}-\frac{1}{4+x}=\frac{11}{30}$$

Solve for x

$x = \frac{\sqrt{18241} - 27}{22} \approx 4.911783921$
$x=\frac{-\sqrt{18241}-27}{22}\approx -7.366329375$

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 $-4,7$ since division by zero is not defined. Multiply both sides of the equation by $30\left(x-7\right)\left(x+4\right)$, the least common multiple of $7-x,4+x,30$.

$$-120-30x-\left(30x-210\right)=11\left(x-7\right)\left(x+4\right)$$

To find the opposite of $30x-210$, find the opposite of each term.

$$-120-30x-30x+210=11\left(x-7\right)\left(x+4\right)$$

Combine $-30x$ and $-30x$ to get $-60x$.

$$-120-60x+210=11\left(x-7\right)\left(x+4\right)$$

Add $-120$ and $210$ to get $90$.

$$90-60x=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply $11$ by $x-7$.

$$90-60x=\left(11x-77\right)\left(x+4\right)$$

Use the distributive property to multiply $11x-77$ by $x+4$ and combine like terms.

$$90-60x=11x^{2}-33x-308$$

Subtract $11x^{2}$ from both sides.

$$90-60x-11x^{2}=-33x-308$$

Add $33x$ to both sides.

$$90-60x-11x^{2}+33x=-308$$

Combine $-60x$ and $33x$ to get $-27x$.

$$90-27x-11x^{2}=-308$$

Add $308$ to both sides.

$$90-27x-11x^{2}+308=0$$

Add $90$ and $308$ to get $398$.

$$398-27x-11x^{2}=0$$

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.

$$-11x^{2}-27x+398=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-11$ for $a$, $-27$ for $b$, and $398$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}-4\left(-11\right)\times 398}}{2\left(-11\right)}$$

Square $-27$.

$$x=\frac{-\left(-27\right)±\sqrt{729-4\left(-11\right)\times 398}}{2\left(-11\right)}$$

Multiply $-4$ times $-11$.

$$x=\frac{-\left(-27\right)±\sqrt{729+44\times 398}}{2\left(-11\right)}$$

Multiply $44$ times $398$.

$$x=\frac{-\left(-27\right)±\sqrt{729+17512}}{2\left(-11\right)}$$

Add $729$ to $17512$.

$$x=\frac{-\left(-27\right)±\sqrt{18241}}{2\left(-11\right)}$$

The opposite of $-27$ is $27$.

$$x=\frac{27±\sqrt{18241}}{2\left(-11\right)}$$

Multiply $2$ times $-11$.

$$x=\frac{27±\sqrt{18241}}{-22}$$

Now solve the equation $x=\frac{27±\sqrt{18241}}{-22}$ when $±$ is plus. Add $27$ to $\sqrt{18241}$.

$$x=\frac{\sqrt{18241}+27}{-22}$$

Divide $27+\sqrt{18241}$ by $-22$.

$$x=\frac{-\sqrt{18241}-27}{22}$$

Now solve the equation $x=\frac{27±\sqrt{18241}}{-22}$ when $±$ is minus. Subtract $\sqrt{18241}$ from $27$.

$$x=\frac{27-\sqrt{18241}}{-22}$$

Divide $27-\sqrt{18241}$ by $-22$.

$$x=\frac{\sqrt{18241}-27}{22}$$

The equation is now solved.

$$x=\frac{-\sqrt{18241}-27}{22}$$ $$x=\frac{\sqrt{18241}-27}{22}$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $-4,7$ since division by zero is not defined. Multiply both sides of the equation by $30\left(x-7\right)\left(x+4\right)$, the least common multiple of $7-x,4+x,30$.

$$-120-30x-\left(30x-210\right)=11\left(x-7\right)\left(x+4\right)$$

To find the opposite of $30x-210$, find the opposite of each term.

$$-120-30x-30x+210=11\left(x-7\right)\left(x+4\right)$$

Combine $-30x$ and $-30x$ to get $-60x$.

$$-120-60x+210=11\left(x-7\right)\left(x+4\right)$$

Add $-120$ and $210$ to get $90$.

$$90-60x=11\left(x-7\right)\left(x+4\right)$$

Use the distributive property to multiply $11$ by $x-7$.

$$90-60x=\left(11x-77\right)\left(x+4\right)$$

Use the distributive property to multiply $11x-77$ by $x+4$ and combine like terms.

$$90-60x=11x^{2}-33x-308$$

Subtract $11x^{2}$ from both sides.

$$90-60x-11x^{2}=-33x-308$$

Add $33x$ to both sides.

$$90-60x-11x^{2}+33x=-308$$

Combine $-60x$ and $33x$ to get $-27x$.

$$90-27x-11x^{2}=-308$$

Subtract $90$ from both sides.

$$-27x-11x^{2}=-308-90$$

Subtract $90$ from $-308$ to get $-398$.

$$-27x-11x^{2}=-398$$

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 $x^{2}+bx=c$.

$$-11x^{2}-27x=-398$$

Divide both sides by $-11$.

$$\frac{-11x^{2}-27x}{-11}=-\frac{398}{-11}$$

Dividing by $-11$ undoes the multiplication by $-11$.

$$x^{2}+\left(-\frac{27}{-11}\right)x=-\frac{398}{-11}$$

Divide $-27$ by $-11$.

$$x^{2}+\frac{27}{11}x=-\frac{398}{-11}$$

Divide $-398$ by $-11$.

$$x^{2}+\frac{27}{11}x=\frac{398}{11}$$

Divide $\frac{27}{11}$, the coefficient of the $x$ term, by $2$ to get $\frac{27}{22}$. Then add the square of $\frac{27}{22}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{27}{11}x+\left(\frac{27}{22}\right)^{2}=\frac{398}{11}+\left(\frac{27}{22}\right)^{2}$$

Square $\frac{27}{22}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{27}{11}x+\frac{729}{484}=\frac{398}{11}+\frac{729}{484}$$

Add $\frac{398}{11}$ to $\frac{729}{484}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{27}{11}x+\frac{729}{484}=\frac{18241}{484}$$

Factor $x^{2}+\frac{27}{11}x+\frac{729}{484}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{27}{22}\right)^{2}=\frac{18241}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{27}{22}\right)^{2}}=\sqrt{\frac{18241}{484}}$$

Simplify.

$$x+\frac{27}{22}=\frac{\sqrt{18241}}{22}$$ $$x+\frac{27}{22}=-\frac{\sqrt{18241}}{22}$$

Subtract $\frac{27}{22}$ from both sides of the equation.

$$x=\frac{\sqrt{18241}-27}{22}$$ $$x=\frac{-\sqrt{18241}-27}{22}$$