Solve (x+a)(x+b)=(x+a)x+(x+a)b

$$( x + a ) ( x + b ) = ( x + a ) x + ( x + a ) b$$

Solve for a (complex solution)

$$a\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply x+a by x+b.

$$x^{2}+xb+ax+ab=\left(x+a\right)x+\left(x+a\right)b$$

Use the distributive property to multiply x+a by x.

$$x^{2}+xb+ax+ab=x^{2}+ax+\left(x+a\right)b$$

Use the distributive property to multiply x+a by b.

$$x^{2}+xb+ax+ab=x^{2}+ax+xb+ab$$

Subtract ax from both sides.

$$x^{2}+xb+ax+ab-ax=x^{2}+xb+ab$$

Combine ax and -ax to get 0.

$$x^{2}+xb+ab=x^{2}+xb+ab$$

Subtract ab from both sides.

$$x^{2}+xb+ab-ab=x^{2}+xb$$

Combine ab and -ab to get 0.

$$x^{2}+xb=x^{2}+xb$$

Reorder the terms.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{C}$$

Solve for b (complex solution)

$$b\in \mathrm{C}$$

Solution Steps

Use the distributive property to multiply x+a by x+b.

$$x^{2}+xb+ax+ab=\left(x+a\right)x+\left(x+a\right)b$$

Use the distributive property to multiply x+a by x.

$$x^{2}+xb+ax+ab=x^{2}+ax+\left(x+a\right)b$$

Use the distributive property to multiply x+a by b.

$$x^{2}+xb+ax+ab=x^{2}+ax+xb+ab$$

Subtract xb from both sides.

$$x^{2}+xb+ax+ab-xb=x^{2}+ax+ab$$

Combine xb and -xb to get 0.

$$x^{2}+ax+ab=x^{2}+ax+ab$$

Subtract ab from both sides.

$$x^{2}+ax+ab-ab=x^{2}+ax$$

Combine ab and -ab to get 0.

$$x^{2}+ax=x^{2}+ax$$

Reorder the terms.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{C}$$

Solve for a

$$a\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply x+a by x+b.

$$x^{2}+xb+ax+ab=\left(x+a\right)x+\left(x+a\right)b$$

Use the distributive property to multiply x+a by x.

$$x^{2}+xb+ax+ab=x^{2}+ax+\left(x+a\right)b$$

Use the distributive property to multiply x+a by b.

$$x^{2}+xb+ax+ab=x^{2}+ax+xb+ab$$

Subtract ax from both sides.

$$x^{2}+xb+ax+ab-ax=x^{2}+xb+ab$$

Combine ax and -ax to get 0.

$$x^{2}+xb+ab=x^{2}+xb+ab$$

Subtract ab from both sides.

$$x^{2}+xb+ab-ab=x^{2}+xb$$

Combine ab and -ab to get 0.

$$x^{2}+xb=x^{2}+xb$$

Reorder the terms.

$$\text{true}$$

This is true for any a.

$$a\in \mathrm{R}$$

Solve for b

$$b\in \mathrm{R}$$

Solution Steps

Use the distributive property to multiply x+a by x+b.

$$x^{2}+xb+ax+ab=\left(x+a\right)x+\left(x+a\right)b$$

Use the distributive property to multiply x+a by x.

$$x^{2}+xb+ax+ab=x^{2}+ax+\left(x+a\right)b$$

Use the distributive property to multiply x+a by b.

$$x^{2}+xb+ax+ab=x^{2}+ax+xb+ab$$

Subtract xb from both sides.

$$x^{2}+xb+ax+ab-xb=x^{2}+ax+ab$$

Combine xb and -xb to get 0.

$$x^{2}+ax+ab=x^{2}+ax+ab$$

Subtract ab from both sides.

$$x^{2}+ax+ab-ab=x^{2}+ax$$

Combine ab and -ab to get 0.

$$x^{2}+ax=x^{2}+ax$$

Reorder the terms.

$$\text{true}$$

This is true for any b.

$$b\in \mathrm{R}$$