Solve (x+9)(x+h)+2=(x+3)^2+k

$$( x + 9 ) ( x + h ) + 2 = ( x + 3 ) ^ { 2 } + k$$

Solve for h (complex solution)

$$\left\{\begin{matrix}h=-\frac{3x-k-7}{x+9}\text{, }&x\neq -9\\h\in \mathrm{C}\text{, }&k=-34\text{ and }x=-9\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+9 by x+h.

$$x^{2}+xh+9x+9h+2=\left(x+3\right)^{2}+k$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+xh+9x+9h+2=x^{2}+6x+9+k$$

Subtract x^{2} from both sides.

$$xh+9x+9h+2=x^{2}+6x+9+k-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$xh+9x+9h+2=6x+9+k$$

Subtract 9x from both sides.

$$xh+9h+2=6x+9+k-9x$$

Combine 6x and -9x to get -3x.

$$xh+9h+2=-3x+9+k$$

Subtract 2 from both sides.

$$xh+9h=-3x+9+k-2$$

Subtract 2 from 9 to get 7.

$$xh+9h=-3x+7+k$$

Combine all terms containing h.

$$\left(x+9\right)h=-3x+7+k$$

The equation is in standard form.

$$\left(x+9\right)h=7+k-3x$$

Divide both sides by x+9.

$$\frac{\left(x+9\right)h}{x+9}=\frac{7+k-3x}{x+9}$$

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

$$h=\frac{7+k-3x}{x+9}$$

Solve for h

$$\left\{\begin{matrix}h=-\frac{3x-k-7}{x+9}\text{, }&x\neq -9\\h\in \mathrm{R}\text{, }&k=-34\text{ and }x=-9\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x+9 by x+h.

$$x^{2}+xh+9x+9h+2=\left(x+3\right)^{2}+k$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+xh+9x+9h+2=x^{2}+6x+9+k$$

Subtract x^{2} from both sides.

$$xh+9x+9h+2=x^{2}+6x+9+k-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$xh+9x+9h+2=6x+9+k$$

Subtract 9x from both sides.

$$xh+9h+2=6x+9+k-9x$$

Combine 6x and -9x to get -3x.

$$xh+9h+2=-3x+9+k$$

Subtract 2 from both sides.

$$xh+9h=-3x+9+k-2$$

Subtract 2 from 9 to get 7.

$$xh+9h=-3x+7+k$$

Combine all terms containing h.

$$\left(x+9\right)h=-3x+7+k$$

The equation is in standard form.

$$\left(x+9\right)h=7+k-3x$$

Divide both sides by x+9.

$$\frac{\left(x+9\right)h}{x+9}=\frac{7+k-3x}{x+9}$$

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

$$h=\frac{7+k-3x}{x+9}$$

Solve for k

$$k=hx+3x+9h-7$$

Solution Steps

Use the distributive property to multiply x+9 by x+h.

$$x^{2}+xh+9x+9h+2=\left(x+3\right)^{2}+k$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$x^{2}+xh+9x+9h+2=x^{2}+6x+9+k$$

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

$$x^{2}+6x+9+k=x^{2}+xh+9x+9h+2$$

Subtract x^{2} from both sides.

$$6x+9+k=x^{2}+xh+9x+9h+2-x^{2}$$

Combine x^{2} and -x^{2} to get 0.

$$6x+9+k=xh+9x+9h+2$$

Subtract 6x from both sides.

$$9+k=xh+9x+9h+2-6x$$

Combine 9x and -6x to get 3x.

$$9+k=xh+3x+9h+2$$

Subtract 9 from both sides.

$$k=xh+3x+9h+2-9$$

Subtract 9 from 2 to get -7.

$$k=xh+3x+9h-7$$