Solve (x + 1)/(x + 2) + (x + 2)/(x + 1) = 25/12 by factorization method) | 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

$$\frac { x + 1 } { x + 2 } + \frac { x + 2 } { x + 1 } = \frac { 25 } { 12 }$$

Solve for x

$x=-5$
$x=2$

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 $-2,-1$ since division by zero is not defined. Multiply both sides of the equation by $12\left(x+1\right)\left(x+2\right)$, the least common multiple of $x+2,x+1,12$.

$$\left(12x+12\right)\left(x+1\right)+\left(12x+24\right)\left(x+2\right)=25\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply $12x+12$ by $x+1$ and combine like terms.

$$12x^{2}+24x+12+\left(12x+24\right)\left(x+2\right)=25\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply $12x+24$ by $x+2$ and combine like terms.

$$12x^{2}+24x+12+12x^{2}+48x+48=25\left(x+1\right)\left(x+2\right)$$

Combine $12x^{2}$ and $12x^{2}$ to get $24x^{2}$.

$$24x^{2}+24x+12+48x+48=25\left(x+1\right)\left(x+2\right)$$

Combine $24x$ and $48x$ to get $72x$.

$$24x^{2}+72x+12+48=25\left(x+1\right)\left(x+2\right)$$

Add $12$ and $48$ to get $60$.

$$24x^{2}+72x+60=25\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply $25$ by $x+1$.

$$24x^{2}+72x+60=\left(25x+25\right)\left(x+2\right)$$

Use the distributive property to multiply $25x+25$ by $x+2$ and combine like terms.

$$24x^{2}+72x+60=25x^{2}+75x+50$$

Subtract $25x^{2}$ from both sides.

$$24x^{2}+72x+60-25x^{2}=75x+50$$

Combine $24x^{2}$ and $-25x^{2}$ to get $-x^{2}$.

$$-x^{2}+72x+60=75x+50$$

Subtract $75x$ from both sides.

$$-x^{2}+72x+60-75x=50$$

Combine $72x$ and $-75x$ to get $-3x$.

$$-x^{2}-3x+60=50$$

Subtract $50$ from both sides.

$$-x^{2}-3x+60-50=0$$

Subtract $50$ from $60$ to get $10$.

$$-x^{2}-3x+10=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1$ for $a$, $-3$ for $b$, and $10$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square $-3$.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply $-4$ times $-1$.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 10}}{2\left(-1\right)}$$

Multiply $4$ times $10$.

$$x=\frac{-\left(-3\right)±\sqrt{9+40}}{2\left(-1\right)}$$

Add $9$ to $40$.

$$x=\frac{-\left(-3\right)±\sqrt{49}}{2\left(-1\right)}$$

Take the square root of $49$.

$$x=\frac{-\left(-3\right)±7}{2\left(-1\right)}$$

The opposite of $-3$ is $3$.

$$x=\frac{3±7}{2\left(-1\right)}$$

Multiply $2$ times $-1$.

$$x=\frac{3±7}{-2}$$

Now solve the equation $x=\frac{3±7}{-2}$ when $±$ is plus. Add $3$ to $7$.

$$x=\frac{10}{-2}$$

Divide $10$ by $-2$.

$$x=-5$$

Now solve the equation $x=\frac{3±7}{-2}$ when $±$ is minus. Subtract $7$ from $3$.

$$x=-\frac{4}{-2}$$

Divide $-4$ by $-2$.

$$x=2$$

The equation is now solved.

$$x=-5$$ $$x=2$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $-2,-1$ since division by zero is not defined. Multiply both sides of the equation by $12\left(x+1\right)\left(x+2\right)$, the least common multiple of $x+2,x+1,12$.

$$\left(12x+12\right)\left(x+1\right)+\left(12x+24\right)\left(x+2\right)=25\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply $12x+12$ by $x+1$ and combine like terms.

$$12x^{2}+24x+12+\left(12x+24\right)\left(x+2\right)=25\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply $12x+24$ by $x+2$ and combine like terms.

$$12x^{2}+24x+12+12x^{2}+48x+48=25\left(x+1\right)\left(x+2\right)$$

Combine $12x^{2}$ and $12x^{2}$ to get $24x^{2}$.

$$24x^{2}+24x+12+48x+48=25\left(x+1\right)\left(x+2\right)$$

Combine $24x$ and $48x$ to get $72x$.

$$24x^{2}+72x+12+48=25\left(x+1\right)\left(x+2\right)$$

Add $12$ and $48$ to get $60$.

$$24x^{2}+72x+60=25\left(x+1\right)\left(x+2\right)$$

Use the distributive property to multiply $25$ by $x+1$.

$$24x^{2}+72x+60=\left(25x+25\right)\left(x+2\right)$$

Use the distributive property to multiply $25x+25$ by $x+2$ and combine like terms.

$$24x^{2}+72x+60=25x^{2}+75x+50$$

Subtract $25x^{2}$ from both sides.

$$24x^{2}+72x+60-25x^{2}=75x+50$$

Combine $24x^{2}$ and $-25x^{2}$ to get $-x^{2}$.

$$-x^{2}+72x+60=75x+50$$

Subtract $75x$ from both sides.

$$-x^{2}+72x+60-75x=50$$

Combine $72x$ and $-75x$ to get $-3x$.

$$-x^{2}-3x+60=50$$

Subtract $60$ from both sides.

$$-x^{2}-3x=50-60$$

Subtract $60$ from $50$ to get $-10$.

$$-x^{2}-3x=-10$$

Divide both sides by $-1$.

$$\frac{-x^{2}-3x}{-1}=-\frac{10}{-1}$$

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

$$x^{2}+\left(-\frac{3}{-1}\right)x=-\frac{10}{-1}$$

Divide $-3$ by $-1$.

$$x^{2}+3x=-\frac{10}{-1}$$

Divide $-10$ by $-1$.

$$x^{2}+3x=10$$

Divide $3$, the coefficient of the $x$ term, by $2$ to get $\frac{3}{2}$. Then add the square of $\frac{3}{2}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Square $\frac{3}{2}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=10+\frac{9}{4}$$

Add $10$ to $\frac{9}{4}$.

$$x^{2}+3x+\frac{9}{4}=\frac{49}{4}$$

Factor $x^{2}+3x+\frac{9}{4}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$x+\frac{3}{2}=\frac{7}{2}$$ $$x+\frac{3}{2}=-\frac{7}{2}$$

Subtract $\frac{3}{2}$ from both sides of the equation.

$$x=2$$ $$x=-5$$