Completing the Square

Completing the Square

Completing the square rewrites a quadratic as a squared bracket plus or minus a constant. For $x^2+bx$, take half of $b$ and square it. This form reveals the turning point and gives a reliable route to solve quadratics that do not factor neatly. If the coefficient of $x^2$ is not 1, first factor it out of the first two terms.

Worked reasoning

$x^2+6x+5=0$ becomes $(x+3)^2-9+5=0$. Thus $(x+3)^2=4$, so $x+3=\pm2$ and $x=-1$ or $x=-5$.

Exam method

1. Move the constant if solving. 2. Add and subtract the square of half the $x$ coefficient. 3. Write a perfect square, take both roots, and check.

Link algebra to the graph

Completed-square form is valuable because it carries geometry in its brackets. In $a(x-p)^2+q$, the graph has turning point $(p,q)$ and axis $x=p$. When solving, keep the equation balanced: add and subtract the same square, or move the constant before making a perfect square. A good self-check expands the bracket once. If the middle term and constant return correctly, the completed-square form is trustworthy.

One-minute retrieval: Completing the Square

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.