Solve for a x^2-12x+a=(x+b)^2

$x^{2} - 12 x + a = {(x + b)}^{2}$

Simplify ${(x + b)}^{2}$ .

Rewrite.

$x^{2} - 12 x + a = 0 + 0 + {(x + b)}^{2}$

Rewrite ${(x + b)}^{2}$ as $(x + b) (x + b)$ .

$x^{2} - 12 x + a = (x + b) (x + b)$

Expand $(x + b) (x + b)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 12 x + a = x (x + b) + b (x + b)$

Apply the distributive property.

$x^{2} - 12 x + a = x \cdot x + x b + b (x + b)$

Apply the distributive property.

$x^{2} - 12 x + a = x \cdot x + x b + b x + b \cdot b$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 12 x + a = x^{2} + x b + b x + b \cdot b$

Multiply $b$ by $b$ .

$x^{2} - 12 x + a = x^{2} + x b + b x + b^{2}$

Add $x b$ and $b x$ .

Reorder $x$ and $b$ .

$x^{2} - 12 x + a = x^{2} + b x + b x + b^{2}$

Add $b x$ and $b x$ .

$x^{2} - 12 x + a = x^{2} + 2 b x + b^{2}$

Move all terms not containing $a$ to the right side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$- 12 x + a = x^{2} + 2 b x + b^{2} - x^{2}$

Add $12 x$ to both sides of the equation.

$a = x^{2} + 2 b x + b^{2} - x^{2} + 12 x$

Combine the opposite terms in $x^{2} + 2 b x + b^{2} - x^{2} + 12 x$ .

Subtract $x^{2}$ from $x^{2}$ .

$a = 2 b x + b^{2} + 0 + 12 x$

Add $2 b x + b^{2}$ and $0$ .

$a = 2 b x + b^{2} + 12 x$