Solve (vi) (x + 1)/(x + 2) + (x + 2)/(x + 3) = 25/12 | 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

$$\quad \frac { x + 1 } { x + 2 } + \frac { x + 2 } { x + 3 } = \frac { 25 } { 12 }$$

Solve for x

$x=\frac{\sqrt{577}-29}{2}\approx -2.489587851$
$x=\frac{-\sqrt{577}-29}{2}\approx -26.510412149$

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 $-3,-2$ since division by zero is not defined. Multiply both sides of the equation by $12\left(x+2\right)\left(x+3\right)$, the least common multiple of $x+2,x+3,12$.

$$\left(12x+36\right)\left(x+1\right)+\left(12x+24\right)\left(x+2\right)=25\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $12x+36$ by $x+1$ and combine like terms.

$$12x^{2}+48x+36+\left(12x+24\right)\left(x+2\right)=25\left(x+2\right)\left(x+3\right)$$

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

$$12x^{2}+48x+36+12x^{2}+48x+48=25\left(x+2\right)\left(x+3\right)$$

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

$$24x^{2}+48x+36+48x+48=25\left(x+2\right)\left(x+3\right)$$

Combine $48x$ and $48x$ to get $96x$.

$$24x^{2}+96x+36+48=25\left(x+2\right)\left(x+3\right)$$

Add $36$ and $48$ to get $84$.

$$24x^{2}+96x+84=25\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $25$ by $x+2$.

$$24x^{2}+96x+84=\left(25x+50\right)\left(x+3\right)$$

Use the distributive property to multiply $25x+50$ by $x+3$ and combine like terms.

$$24x^{2}+96x+84=25x^{2}+125x+150$$

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

$$24x^{2}+96x+84-25x^{2}=125x+150$$

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

$$-x^{2}+96x+84=125x+150$$

Subtract $125x$ from both sides.

$$-x^{2}+96x+84-125x=150$$

Combine $96x$ and $-125x$ to get $-29x$.

$$-x^{2}-29x+84=150$$

Subtract $150$ from both sides.

$$-x^{2}-29x+84-150=0$$

Subtract $150$ from $84$ to get $-66$.

$$-x^{2}-29x-66=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-1$ for $a$, $-29$ for $b$, and $-66$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\left(-1\right)\left(-66\right)}}{2\left(-1\right)}$$

Square $-29$.

$$x=\frac{-\left(-29\right)±\sqrt{841-4\left(-1\right)\left(-66\right)}}{2\left(-1\right)}$$

Multiply $-4$ times $-1$.

$$x=\frac{-\left(-29\right)±\sqrt{841+4\left(-66\right)}}{2\left(-1\right)}$$

Multiply $4$ times $-66$.

$$x=\frac{-\left(-29\right)±\sqrt{841-264}}{2\left(-1\right)}$$

Add $841$ to $-264$.

$$x=\frac{-\left(-29\right)±\sqrt{577}}{2\left(-1\right)}$$

The opposite of $-29$ is $29$.

$$x=\frac{29±\sqrt{577}}{2\left(-1\right)}$$

Multiply $2$ times $-1$.

$$x=\frac{29±\sqrt{577}}{-2}$$

Now solve the equation $x=\frac{29±\sqrt{577}}{-2}$ when $±$ is plus. Add $29$ to $\sqrt{577}$.

$$x=\frac{\sqrt{577}+29}{-2}$$

Divide $29+\sqrt{577}$ by $-2$.

$$x=\frac{-\sqrt{577}-29}{2}$$

Now solve the equation $x=\frac{29±\sqrt{577}}{-2}$ when $±$ is minus. Subtract $\sqrt{577}$ from $29$.

$$x=\frac{29-\sqrt{577}}{-2}$$

Divide $29-\sqrt{577}$ by $-2$.

$$x=\frac{\sqrt{577}-29}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{577}-29}{2}$$ $$x=\frac{\sqrt{577}-29}{2}$$

Steps for Completing the Square

Variable $x$ cannot be equal to any of the values $-3,-2$ since division by zero is not defined. Multiply both sides of the equation by $12\left(x+2\right)\left(x+3\right)$, the least common multiple of $x+2,x+3,12$.

$$\left(12x+36\right)\left(x+1\right)+\left(12x+24\right)\left(x+2\right)=25\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $12x+36$ by $x+1$ and combine like terms.

$$12x^{2}+48x+36+\left(12x+24\right)\left(x+2\right)=25\left(x+2\right)\left(x+3\right)$$

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

$$12x^{2}+48x+36+12x^{2}+48x+48=25\left(x+2\right)\left(x+3\right)$$

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

$$24x^{2}+48x+36+48x+48=25\left(x+2\right)\left(x+3\right)$$

Combine $48x$ and $48x$ to get $96x$.

$$24x^{2}+96x+36+48=25\left(x+2\right)\left(x+3\right)$$

Add $36$ and $48$ to get $84$.

$$24x^{2}+96x+84=25\left(x+2\right)\left(x+3\right)$$

Use the distributive property to multiply $25$ by $x+2$.

$$24x^{2}+96x+84=\left(25x+50\right)\left(x+3\right)$$

Use the distributive property to multiply $25x+50$ by $x+3$ and combine like terms.

$$24x^{2}+96x+84=25x^{2}+125x+150$$

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

$$24x^{2}+96x+84-25x^{2}=125x+150$$

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

$$-x^{2}+96x+84=125x+150$$

Subtract $125x$ from both sides.

$$-x^{2}+96x+84-125x=150$$

Combine $96x$ and $-125x$ to get $-29x$.

$$-x^{2}-29x+84=150$$

Subtract $84$ from both sides.

$$-x^{2}-29x=150-84$$

Subtract $84$ from $150$ to get $66$.

$$-x^{2}-29x=66$$

Divide both sides by $-1$.

$$\frac{-x^{2}-29x}{-1}=\frac{66}{-1}$$

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

$$x^{2}+\left(-\frac{29}{-1}\right)x=\frac{66}{-1}$$

Divide $-29$ by $-1$.

$$x^{2}+29x=\frac{66}{-1}$$

Divide $66$ by $-1$.

$$x^{2}+29x=-66$$

Divide $29$, the coefficient of the $x$ term, by $2$ to get $\frac{29}{2}$. Then add the square of $\frac{29}{2}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+29x+\left(\frac{29}{2}\right)^{2}=-66+\left(\frac{29}{2}\right)^{2}$$

Square $\frac{29}{2}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+29x+\frac{841}{4}=-66+\frac{841}{4}$$

Add $-66$ to $\frac{841}{4}$.

$$x^{2}+29x+\frac{841}{4}=\frac{577}{4}$$

Factor $x^{2}+29x+\frac{841}{4}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{29}{2}\right)^{2}=\frac{577}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{29}{2}\right)^{2}}=\sqrt{\frac{577}{4}}$$

Simplify.

$$x+\frac{29}{2}=\frac{\sqrt{577}}{2}$$ $$x+\frac{29}{2}=-\frac{\sqrt{577}}{2}$$

Subtract $\frac{29}{2}$ from both sides of the equation.

$$x=\frac{\sqrt{577}-29}{2}$$ $$x=\frac{-\sqrt{577}-29}{2}$$