Solve 8x^2+36+47=2Ax^2+7Ax+6A+6Bx+9B+3x^2+5x-2

$$8 x ^ { 2 } + 36 + 47 = 2 A x ^ { 2 } + 7 A x + 6 A + 6 B x + 9 B + 3 x ^ { 2 } + 5 x - 2$$

Solve for A (complex solution)

$$\left\{\begin{matrix}A=-\frac{-5x^{2}+6Bx+5x+9B-85}{\left(x+2\right)\left(2x+3\right)}\text{, }&x\neq -\frac{3}{2}\text{ and }x\neq -2\\A\in \mathrm{C}\text{, }&x=-2\text{ and }B=-\frac{115}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 36 and 47 to get 83.

$$8x^{2}+83=2Ax^{2}+7Ax+6A+6Bx+9B+3x^{2}+5x-2$$

Swap sides so that all variable terms are on the left hand side.

$$2Ax^{2}+7Ax+6A+6Bx+9B+3x^{2}+5x-2=8x^{2}+83$$

Subtract 6Bx from both sides.

$$2Ax^{2}+7Ax+6A+9B+3x^{2}+5x-2=8x^{2}+83-6Bx$$

Subtract 9B from both sides.

$$2Ax^{2}+7Ax+6A+3x^{2}+5x-2=8x^{2}+83-6Bx-9B$$

Subtract 3x^{2} from both sides.

$$2Ax^{2}+7Ax+6A+5x-2=8x^{2}+83-6Bx-9B-3x^{2}$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$2Ax^{2}+7Ax+6A+5x-2=5x^{2}+83-6Bx-9B$$

Subtract 5x from both sides.

$$2Ax^{2}+7Ax+6A-2=5x^{2}+83-6Bx-9B-5x$$

Add 2 to both sides.

$$2Ax^{2}+7Ax+6A=5x^{2}+83-6Bx-9B-5x+2$$

Add 83 and 2 to get 85.

$$2Ax^{2}+7Ax+6A=5x^{2}+85-6Bx-9B-5x$$

Combine all terms containing A.

$$\left(2x^{2}+7x+6\right)A=5x^{2}+85-6Bx-9B-5x$$

The equation is in standard form.

$$\left(2x^{2}+7x+6\right)A=5x^{2}-6Bx-5x-9B+85$$

Divide both sides by 2x^{2}+7x+6.

$$\frac{\left(2x^{2}+7x+6\right)A}{2x^{2}+7x+6}=\frac{5x^{2}-6Bx-5x-9B+85}{2x^{2}+7x+6}$$

Dividing by 2x^{2}+7x+6 undoes the multiplication by 2x^{2}+7x+6.

$$A=\frac{5x^{2}-6Bx-5x-9B+85}{2x^{2}+7x+6}$$

Divide 5x^{2}+85-6Bx-9B-5x by 2x^{2}+7x+6.

$$A=\frac{5x^{2}-6Bx-5x-9B+85}{\left(x+2\right)\left(2x+3\right)}$$

Solve for A

$$\left\{\begin{matrix}A=-\frac{-5x^{2}+6Bx+5x+9B-85}{\left(x+2\right)\left(2x+3\right)}\text{, }&x\neq -\frac{3}{2}\text{ and }x\neq -2\\A\in \mathrm{R}\text{, }&x=-2\text{ and }B=-\frac{115}{3}\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 36 and 47 to get 83.

$$8x^{2}+83=2Ax^{2}+7Ax+6A+6Bx+9B+3x^{2}+5x-2$$

Swap sides so that all variable terms are on the left hand side.

$$2Ax^{2}+7Ax+6A+6Bx+9B+3x^{2}+5x-2=8x^{2}+83$$

Subtract 6Bx from both sides.

$$2Ax^{2}+7Ax+6A+9B+3x^{2}+5x-2=8x^{2}+83-6Bx$$

Subtract 9B from both sides.

$$2Ax^{2}+7Ax+6A+3x^{2}+5x-2=8x^{2}+83-6Bx-9B$$

Subtract 3x^{2} from both sides.

$$2Ax^{2}+7Ax+6A+5x-2=8x^{2}+83-6Bx-9B-3x^{2}$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$2Ax^{2}+7Ax+6A+5x-2=5x^{2}+83-6Bx-9B$$

Subtract 5x from both sides.

$$2Ax^{2}+7Ax+6A-2=5x^{2}+83-6Bx-9B-5x$$

Add 2 to both sides.

$$2Ax^{2}+7Ax+6A=5x^{2}+83-6Bx-9B-5x+2$$

Add 83 and 2 to get 85.

$$2Ax^{2}+7Ax+6A=5x^{2}+85-6Bx-9B-5x$$

Combine all terms containing A.

$$\left(2x^{2}+7x+6\right)A=5x^{2}+85-6Bx-9B-5x$$

The equation is in standard form.

$$\left(2x^{2}+7x+6\right)A=5x^{2}-6Bx-5x-9B+85$$

Divide both sides by 2x^{2}+7x+6.

$$\frac{\left(2x^{2}+7x+6\right)A}{2x^{2}+7x+6}=\frac{5x^{2}-6Bx-5x-9B+85}{2x^{2}+7x+6}$$

Dividing by 2x^{2}+7x+6 undoes the multiplication by 2x^{2}+7x+6.

$$A=\frac{5x^{2}-6Bx-5x-9B+85}{2x^{2}+7x+6}$$

Divide 5x^{2}+85-6Bx-9B-5x by 2x^{2}+7x+6.

$$A=\frac{5x^{2}-6Bx-5x-9B+85}{\left(x+2\right)\left(2x+3\right)}$$

Solve for B

$$B=-\frac{2Ax^{2}-5x^{2}+7Ax+5x+6A-85}{3\left(2x+3\right)}$$

$$x\neq -\frac{3}{2}$$

Steps for Solving Linear Equation

Add 36 and 47 to get 83.

$$8x^{2}+83=2Ax^{2}+7Ax+6A+6Bx+9B+3x^{2}+5x-2$$

Swap sides so that all variable terms are on the left hand side.

$$2Ax^{2}+7Ax+6A+6Bx+9B+3x^{2}+5x-2=8x^{2}+83$$

Subtract 2Ax^{2} from both sides.

$$7Ax+6A+6Bx+9B+3x^{2}+5x-2=8x^{2}+83-2Ax^{2}$$

Subtract 7Ax from both sides.

$$6A+6Bx+9B+3x^{2}+5x-2=8x^{2}+83-2Ax^{2}-7Ax$$

Subtract 6A from both sides.

$$6Bx+9B+3x^{2}+5x-2=8x^{2}+83-2Ax^{2}-7Ax-6A$$

Subtract 3x^{2} from both sides.

$$6Bx+9B+5x-2=8x^{2}+83-2Ax^{2}-7Ax-6A-3x^{2}$$

Subtract 5x from both sides.

$$6Bx+9B-2=8x^{2}+83-2Ax^{2}-7Ax-6A-3x^{2}-5x$$

Add 2 to both sides.

$$6Bx+9B=8x^{2}+83-2Ax^{2}-7Ax-6A-3x^{2}-5x+2$$

Combine 8x^{2} and -3x^{2} to get 5x^{2}.

$$6Bx+9B=5x^{2}+83-2Ax^{2}-7Ax-6A-5x+2$$

Add 83 and 2 to get 85.

$$6Bx+9B=5x^{2}+85-2Ax^{2}-7Ax-6A-5x$$

Combine all terms containing B.

$$\left(6x+9\right)B=5x^{2}+85-2Ax^{2}-7Ax-6A-5x$$

The equation is in standard form.

$$\left(6x+9\right)B=85-6A-5x-7Ax+5x^{2}-2Ax^{2}$$

Divide both sides by 6x+9.

$$\frac{\left(6x+9\right)B}{6x+9}=\frac{85-6A-5x-7Ax+5x^{2}-2Ax^{2}}{6x+9}$$

Dividing by 6x+9 undoes the multiplication by 6x+9.

$$B=\frac{85-6A-5x-7Ax+5x^{2}-2Ax^{2}}{6x+9}$$

Divide 5x^{2}+85-2Ax^{2}-7Ax-6A-5x by 6x+9.

$$B=\frac{85-6A-5x-7Ax+5x^{2}-2Ax^{2}}{3\left(2x+3\right)}$$