Solve (1+A)(x)^(2)+(1+B)x+A+B=0 | 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

$$(1+A) { x }^{ 2 } +(1+B)x+A+B=0$$

Solve for A (complex solution)

$\left\{\begin{matrix}A=-\frac{\left(x+1\right)\left(x+B\right)}{x^{2}+1}\text{, }&x\neq -i\text{ and }x\neq i\\A\in \mathrm{C}\text{, }&\left(x=i\text{ and }B=-i\right)\text{ or }\left(x=-i\text{ and }B=i\right)\end{matrix}\right.$

Steps for Solving Linear Equation

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

$$x^{2}+Ax^{2}+\left(1+B\right)x+A+B=0$$

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

$$x^{2}+Ax^{2}+x+Bx+A+B=0$$

Subtract $x^{2}$ from both sides. Anything subtracted from zero gives its negation.

$$Ax^{2}+x+Bx+A+B=-x^{2}$$

Subtract $x$ from both sides.

$$Ax^{2}+Bx+A+B=-x^{2}-x$$

Subtract $Bx$ from both sides.

$$Ax^{2}+A+B=-x^{2}-x-Bx$$

Subtract $B$ from both sides.

$$Ax^{2}+A=-x^{2}-x-Bx-B$$

Combine all terms containing $A$.

$$\left(x^{2}+1\right)A=-x^{2}-x-Bx-B$$

The equation is in standard form.

$$\left(x^{2}+1\right)A=-x^{2}-Bx-x-B$$

Divide both sides by $x^{2}+1$.

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

Dividing by $x^{2}+1$ undoes the multiplication by $x^{2}+1$.

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

Solve for B (complex solution)

$\left\{\begin{matrix}B=-\frac{Ax^{2}+x^{2}+x+A}{x+1}\text{, }&x\neq -1\\B\in \mathrm{C}\text{, }&A=0\text{ and }x=-1\end{matrix}\right.$

Steps for Solving Linear Equation

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

$$x^{2}+Ax^{2}+\left(1+B\right)x+A+B=0$$

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

$$x^{2}+Ax^{2}+x+Bx+A+B=0$$

Subtract $x^{2}$ from both sides. Anything subtracted from zero gives its negation.

$$Ax^{2}+x+Bx+A+B=-x^{2}$$

Subtract $Ax^{2}$ from both sides.

$$x+Bx+A+B=-x^{2}-Ax^{2}$$

Subtract $x$ from both sides.

$$Bx+A+B=-x^{2}-Ax^{2}-x$$

Subtract $A$ from both sides.

$$Bx+B=-x^{2}-Ax^{2}-x-A$$

Reorder the terms.

$$Bx+B=-Ax^{2}-x^{2}-x-A$$

Combine all terms containing $B$.

$$\left(x+1\right)B=-Ax^{2}-x^{2}-x-A$$

Divide both sides by $1+x$.

$$\frac{\left(x+1\right)B}{x+1}=\frac{-Ax^{2}-x^{2}-x-A}{x+1}$$

Dividing by $1+x$ undoes the multiplication by $1+x$.

$$B=\frac{-Ax^{2}-x^{2}-x-A}{x+1}$$

Divide $-Ax^{2}-x^{2}-x-A$ by $1+x$.

$$B=-\frac{Ax^{2}+x^{2}+x+A}{x+1}$$

Solve for A

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

Steps for Solving Linear Equation

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

$$x^{2}+Ax^{2}+\left(1+B\right)x+A+B=0$$

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

$$x^{2}+Ax^{2}+x+Bx+A+B=0$$

Subtract $x^{2}$ from both sides. Anything subtracted from zero gives its negation.

$$Ax^{2}+x+Bx+A+B=-x^{2}$$

Subtract $x$ from both sides.

$$Ax^{2}+Bx+A+B=-x^{2}-x$$

Subtract $Bx$ from both sides.

$$Ax^{2}+A+B=-x^{2}-x-Bx$$

Subtract $B$ from both sides.

$$Ax^{2}+A=-x^{2}-x-Bx-B$$

Combine all terms containing $A$.

$$\left(x^{2}+1\right)A=-x^{2}-x-Bx-B$$

The equation is in standard form.

$$\left(x^{2}+1\right)A=-x^{2}-Bx-x-B$$

Divide both sides by $x^{2}+1$.

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

Dividing by $x^{2}+1$ undoes the multiplication by $x^{2}+1$.

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

Solve for B

$\left\{\begin{matrix}B=-\frac{Ax^{2}+x^{2}+x+A}{x+1}\text{, }&x\neq -1\\B\in \mathrm{R}\text{, }&A=0\text{ and }x=-1\end{matrix}\right.$

Steps for Solving Linear Equation

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

$$x^{2}+Ax^{2}+\left(1+B\right)x+A+B=0$$

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

$$x^{2}+Ax^{2}+x+Bx+A+B=0$$

Subtract $x^{2}$ from both sides. Anything subtracted from zero gives its negation.

$$Ax^{2}+x+Bx+A+B=-x^{2}$$

Subtract $Ax^{2}$ from both sides.

$$x+Bx+A+B=-x^{2}-Ax^{2}$$

Subtract $x$ from both sides.

$$Bx+A+B=-x^{2}-Ax^{2}-x$$

Subtract $A$ from both sides.

$$Bx+B=-x^{2}-Ax^{2}-x-A$$

Reorder the terms.

$$Bx+B=-Ax^{2}-x^{2}-x-A$$

Combine all terms containing $B$.

$$\left(x+1\right)B=-Ax^{2}-x^{2}-x-A$$

Divide both sides by $x+1$.

$$\frac{\left(x+1\right)B}{x+1}=\frac{-Ax^{2}-x^{2}-x-A}{x+1}$$

Dividing by $x+1$ undoes the multiplication by $x+1$.

$$B=\frac{-Ax^{2}-x^{2}-x-A}{x+1}$$

Divide $-Ax^{2}-x^{2}-x-A$ by $x+1$.

$$B=-\frac{Ax^{2}+x^{2}+x+A}{x+1}$$