Solve (x-a)(x-b)=x^2-(a+b)x+ab

$$( x - a ) ( x - b ) = x ^ { 2 } - ( a + b ) x + a b$$

Solve for a (complex solution)

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

Solution Steps

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

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

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

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

To find the opposite of ax+bx, find the opposite of each term.

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

Add ax to both sides.

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

Combine -ax and ax to get 0.

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

Subtract ab from both sides.

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

Combine ab and -ab to get 0.

$$x^{2}-xb=x^{2}-bx$$

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.

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

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

$$x^{2}-xb-ax+ba=x^{2}-\left(ax+bx\right)+ab$$

To find the opposite of ax+bx, find the opposite of each term.

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

Add bx to both sides.

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

Combine -xb and bx to get 0.

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

Subtract ab from both sides.

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

Combine ba 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.

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

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

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

To find the opposite of ax+bx, find the opposite of each term.

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

Add ax to both sides.

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

Combine -ax and ax to get 0.

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

Subtract ab from both sides.

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

Combine ab and -ab to get 0.

$$x^{2}-xb=x^{2}-bx$$

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.

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

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

$$x^{2}-xb-ax+ba=x^{2}-\left(ax+bx\right)+ab$$

To find the opposite of ax+bx, find the opposite of each term.

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

Add bx to both sides.

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

Combine -xb and bx to get 0.

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

Subtract ab from both sides.

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

Combine ba and -ab to get 0.

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

Reorder the terms.

$$\text{true}$$

This is true for any b.

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