Solving by Completing the Square

Completing the square is the process of rewriting a quadratic expression of the form $x^2 + bx + c$ as $(x + p)^2 + q$, where $p$ is half of the coefficient of $x$. This rewritten form makes the structure of the parabola clearer and allows direct solving by using square roots.

The value of $p$ is always $\frac{b}{2}$ because expanding $(x + p)^2$ gives $x^2 + 2px + p^2$, and matching the middle term requires $2p = b$. This ensures the rewritten expression remains algebraically equivalent to the original quadratic.

The constant $q$ emerges from adjusting for the introduction of $p^2$ during the process. Subtracting or adding the necessary amount ensures the original constant term $c$ is preserved, completing the transformation into a solvable form.

Standard procedure involves isolating $x^2 + bx$, forming $(x + p)^2$ using $p = \frac{b}{2}$, adjusting the constant term, and solving using square roots. This predictable sequence reduces errors and streamlines the solving process.

Handling equations with a leading coefficient requires dividing the entire equation by $a$ when given $ax^2 + bx + c = 0$. This step simplifies the quadratic to one with a leading coefficient of 1, making the square-completion process straightforward.

Solving the resulting equation involves taking $\pm \sqrt{}$ after isolating the squared bracket. This is the final stage where both solutions emerge, and it should be executed carefully to avoid sign errors.

Difference from the quadratic formula: Completing the square is a process, whereas the quadratic formula is a direct result derived from applying this method in general form. Completing the square is more conceptual, while the formula is faster once memorised.

Difference from rearranging and isolating $x$: Completing the square is essential when both $x^2$ and $x$ terms are present. Simple rearrangement works only for equations missing the $x$ term, such as $x^2 = k$.

Check whether to divide first when the quadratic has a leading coefficient other than 1. Dividing simplifies the process, but this step is appropriate only when solving an equation rather than rewriting an expression.

Always include the $\pm$ symbol when taking square roots. Omitting it leads to losing one solution, a common exam mistake that results in lost marks.

Avoid expanding squared brackets unnecessarily because it introduces extra work and increases error risk. The goal is to isolate the squared expression, not undo it.

Forgetting to balance the constant term leads to incorrect equations. After forming $(x + p)^2$, failing to subtract $p^2$ causes the transformed expression to mismatch the original.

Dropping the negative sign when taking square roots is a frequent error. Remembering that $\sqrt{k}$ has two branches is essential for capturing both real solutions.

Dividing incorrectly when dealing with leading coefficients can distort the equation. Every term, including the constant on the right side, must be divided by the same value.

Link to the quadratic formula comes from completing the square on the general quadratic $ax^2 + bx + c = 0$. This derivation reveals why the formula has the structure it does and why the discriminant appears under the radical.

Graphical interpretation shows that completing the square produces the vertex form of a parabola. This directly identifies the vertex $( -p, q )$ and clarifies transformations such as shifts and reflections.

Applications beyond solving equations include rewriting functions to analyze maximum/minimum values, derive motion formulas, and study optimisation problems across mathematics and physics.