Solve 18x+17 (x^ 2 +1)(x+2) = Ax+8 (x^ 2 +0 + c (x+2) | 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

$$\frac{18x+17}{(x^{2}+1)(x+2)}=\frac{Ax+8}{(x^{2}+0}+\frac{c}{(x+2)}$$

Solve for x

$x=\frac{\sqrt{22285}-155}{174}\approx -0.032864293$
$x=\frac{-\sqrt{22285}-155}{174}\approx -1.748744903$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

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

$$15x+17=\left(x+2\right)\left(81x+8\right)+\left(x^{2}+1\right)\times 6$$

Use the distributive property to multiply $x+2$ by $81x+8$ and combine like terms.

$$15x+17=81x^{2}+170x+16+\left(x^{2}+1\right)\times 6$$

Use the distributive property to multiply $x^{2}+1$ by $6$.

$$15x+17=81x^{2}+170x+16+6x^{2}+6$$

Combine $81x^{2}$ and $6x^{2}$ to get $87x^{2}$.

$$15x+17=87x^{2}+170x+16+6$$

Add $16$ and $6$ to get $22$.

$$15x+17=87x^{2}+170x+22$$

Subtract $87x^{2}$ from both sides.

$$15x+17-87x^{2}=170x+22$$

Subtract $170x$ from both sides.

$$15x+17-87x^{2}-170x=22$$

Combine $15x$ and $-170x$ to get $-155x$.

$$-155x+17-87x^{2}=22$$

Subtract $22$ from both sides.

$$-155x+17-87x^{2}-22=0$$

Subtract $22$ from $17$ to get $-5$.

$$-155x-5-87x^{2}=0$$

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.

$$-87x^{2}-155x-5=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-87$ for $a$, $-155$ for $b$, and $-5$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$x=\frac{-\left(-155\right)±\sqrt{\left(-155\right)^{2}-4\left(-87\right)\left(-5\right)}}{2\left(-87\right)}$$

Square $-155$.

$$x=\frac{-\left(-155\right)±\sqrt{24025-4\left(-87\right)\left(-5\right)}}{2\left(-87\right)}$$

Multiply $-4$ times $-87$.

$$x=\frac{-\left(-155\right)±\sqrt{24025+348\left(-5\right)}}{2\left(-87\right)}$$

Multiply $348$ times $-5$.

$$x=\frac{-\left(-155\right)±\sqrt{24025-1740}}{2\left(-87\right)}$$

Add $24025$ to $-1740$.

$$x=\frac{-\left(-155\right)±\sqrt{22285}}{2\left(-87\right)}$$

The opposite of $-155$ is $155$.

$$x=\frac{155±\sqrt{22285}}{2\left(-87\right)}$$

Multiply $2$ times $-87$.

$$x=\frac{155±\sqrt{22285}}{-174}$$

Now solve the equation $x=\frac{155±\sqrt{22285}}{-174}$ when $±$ is plus. Add $155$ to $\sqrt{22285}$.

$$x=\frac{\sqrt{22285}+155}{-174}$$

Divide $155+\sqrt{22285}$ by $-174$.

$$x=\frac{-\sqrt{22285}-155}{174}$$

Now solve the equation $x=\frac{155±\sqrt{22285}}{-174}$ when $±$ is minus. Subtract $\sqrt{22285}$ from $155$.

$$x=\frac{155-\sqrt{22285}}{-174}$$

Divide $155-\sqrt{22285}$ by $-174$.

$$x=\frac{\sqrt{22285}-155}{174}$$

The equation is now solved.

$$x=\frac{-\sqrt{22285}-155}{174}$$ $$x=\frac{\sqrt{22285}-155}{174}$$

Steps for Completing the Square

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

$$15x+17=\left(x+2\right)\left(81x+8\right)+\left(x^{2}+1\right)\times 6$$

Use the distributive property to multiply $x+2$ by $81x+8$ and combine like terms.

$$15x+17=81x^{2}+170x+16+\left(x^{2}+1\right)\times 6$$

Use the distributive property to multiply $x^{2}+1$ by $6$.

$$15x+17=81x^{2}+170x+16+6x^{2}+6$$

Combine $81x^{2}$ and $6x^{2}$ to get $87x^{2}$.

$$15x+17=87x^{2}+170x+16+6$$

Add $16$ and $6$ to get $22$.

$$15x+17=87x^{2}+170x+22$$

Subtract $87x^{2}$ from both sides.

$$15x+17-87x^{2}=170x+22$$

Subtract $170x$ from both sides.

$$15x+17-87x^{2}-170x=22$$

Combine $15x$ and $-170x$ to get $-155x$.

$$-155x+17-87x^{2}=22$$

Subtract $17$ from both sides.

$$-155x-87x^{2}=22-17$$

Subtract $17$ from $22$ to get $5$.

$$-155x-87x^{2}=5$$

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

$$-87x^{2}-155x=5$$

Divide both sides by $-87$.

$$\frac{-87x^{2}-155x}{-87}=\frac{5}{-87}$$

Dividing by $-87$ undoes the multiplication by $-87$.

$$x^{2}+\left(-\frac{155}{-87}\right)x=\frac{5}{-87}$$

Divide $-155$ by $-87$.

$$x^{2}+\frac{155}{87}x=\frac{5}{-87}$$

Divide $5$ by $-87$.

$$x^{2}+\frac{155}{87}x=-\frac{5}{87}$$

Divide $\frac{155}{87}$, the coefficient of the $x$ term, by $2$ to get $\frac{155}{174}$. Then add the square of $\frac{155}{174}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{155}{87}x+\left(\frac{155}{174}\right)^{2}=-\frac{5}{87}+\left(\frac{155}{174}\right)^{2}$$

Square $\frac{155}{174}$ by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{155}{87}x+\frac{24025}{30276}=-\frac{5}{87}+\frac{24025}{30276}$$

Add $-\frac{5}{87}$ to $\frac{24025}{30276}$ by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{155}{87}x+\frac{24025}{30276}=\frac{22285}{30276}$$

Factor $x^{2}+\frac{155}{87}x+\frac{24025}{30276}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(x+\frac{155}{174}\right)^{2}=\frac{22285}{30276}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{155}{174}\right)^{2}}=\sqrt{\frac{22285}{30276}}$$

Simplify.

$$x+\frac{155}{174}=\frac{\sqrt{22285}}{174}$$ $$x+\frac{155}{174}=-\frac{\sqrt{22285}}{174}$$

Subtract $\frac{155}{174}$ from both sides of the equation.

$$x=\frac{\sqrt{22285}-155}{174}$$ $$x=\frac{-\sqrt{22285}-155}{174}$$