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Solve for B
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x=x^{3}+ax^{2}-\left(\beta x-2ax\right)+3a+4B
Use the distributive property to multiply \beta -2a by x.
x=x^{3}+ax^{2}-\beta x+2ax+3a+4B
To find the opposite of \beta x-2ax, find the opposite of each term.
x^{3}+ax^{2}-\beta x+2ax+3a+4B=x
Swap sides so that all variable terms are on the left hand side.
ax^{2}-\beta x+2ax+3a+4B=x-x^{3}
Subtract x^{3} from both sides.
-\beta x+2ax+3a+4B=x-x^{3}-ax^{2}
Subtract ax^{2} from both sides.
2ax+3a+4B=x-x^{3}-ax^{2}+\beta x
Add \beta x to both sides.
3a+4B=x-x^{3}-ax^{2}+\beta x-2ax
Subtract 2ax from both sides.
4B=x-x^{3}-ax^{2}+\beta x-2ax-3a
Subtract 3a from both sides.
4B=-x^{3}-ax^{2}+x\beta -2ax+x-3a
Reorder the terms.
\frac{4B}{4}=\frac{-x^{3}-ax^{2}+x\beta -2ax+x-3a}{4}
Divide both sides by 4.
B=\frac{-x^{3}-ax^{2}+x\beta -2ax+x-3a}{4}
Dividing by 4 undoes the multiplication by 4.
B=-\frac{ax^{2}}{4}+\frac{x\beta }{4}-\frac{ax}{2}-\frac{x^{3}}{4}+\frac{x}{4}-\frac{3a}{4}
Divide -x^{3}-ax^{2}+x\beta -2ax+x-3a by 4.
x=x^{3}+ax^{2}-\left(\beta x-2ax\right)+3a+4B
Use the distributive property to multiply \beta -2a by x.
x=x^{3}+ax^{2}-\beta x+2ax+3a+4B
To find the opposite of \beta x-2ax, find the opposite of each term.
x^{3}+ax^{2}-\beta x+2ax+3a+4B=x
Swap sides so that all variable terms are on the left hand side.
ax^{2}-\beta x+2ax+3a+4B=x-x^{3}
Subtract x^{3} from both sides.
ax^{2}+2ax+3a+4B=x-x^{3}+\beta x
Add \beta x to both sides.
ax^{2}+2ax+3a=x-x^{3}+\beta x-4B
Subtract 4B from both sides.
\left(x^{2}+2x+3\right)a=x-x^{3}+\beta x-4B
Combine all terms containing a.
\left(x^{2}+2x+3\right)a=-x^{3}+x\beta +x-4B
The equation is in standard form.
\frac{\left(x^{2}+2x+3\right)a}{x^{2}+2x+3}=\frac{-x^{3}+x\beta +x-4B}{x^{2}+2x+3}
Divide both sides by x^{2}+2x+3.
a=\frac{-x^{3}+x\beta +x-4B}{x^{2}+2x+3}
Dividing by x^{2}+2x+3 undoes the multiplication by x^{2}+2x+3.