When deriving the quadratic formula by completing the square on $a{x}^2+bx+c=0$, which step justifies introducing the term ${\left(\frac{b}{2a}\right)}^2$ into the equation?

Multiplying both sides by $2a$ and then adding $b^2$ eliminates fractions and directly gives the vertex without completing the square, which is why ${\left(\frac{b}{2a}\right)}^2$ is unnecessary for the derivation.

Subtracting ${\left(\frac{b}{2a}\right)}^2$ from both sides simplifies the expression so the quadratic becomes linear, which directly yields the formula when solved for $x$.

Adding ${\left(\frac{b}{2a}\right)}^2$ to both sides after dividing by $a$ completes the square inside the parentheses and converts $x^2+\left(\frac{b}{a}\right)x$ into ${(x+\frac{b}{2a})}^2$, which is the key algebraic step that produces a perfect square on the left-hand side of the formula.

Replacing $c$ with ${\left(\frac{b}{2a}\right)}^2$ always reduces the discriminant to zero and therefore produces the formula for repeated roots, not the general quadratic formula for all roots.