Completing the square
Completing the square is an algebraic technique used to rewrite a quadratic polynomial in a form that reveals its structural properties. Consider a polynomial of the form:
\[p ( x ) = a x^{2} + b x + c , a \neq 0\]The objective is to determine real constants $h$ and $k$, which depend on $a$, $b$, and $c$, so that $p ( x )$ assumes the vertex form
\[p ( x ) = a ( x + h )^{2} + k\]The pair $( - h , k )$ specifies the vertex of the corresponding parabola. The value $k$ represents the minimum of $p$ when $a > 0$ and the maximum when $a < 0$. Setting $p ( x )$ equal to zero yields the equation $a ( x + h )^{2} = - k$, from which the roots can be obtained by taking square roots of both sides.
To derive explicit expressions for $h$ and $k$, the process begins by factoring $a$ from the quadratic and linear terms:
\[p ( x ) = a ( x^{2} + \frac{b}{a} x ) + c\]The crucial step is to add and subtract $(( \frac{b}{2 a} ))^{ 2}$ inside the parentheses, a quantity chosen so that the three terms involving $x$ form a perfect square trinomial:
\[p ( x ) = a ( x^{2} + \frac{b}{a} x + (( \frac{b}{2 a} ))^{ 2} - (( \frac{b}{2 a} ))^{ 2} ) + c\]Recognising the perfect square trinomial allows it to be rewritten in compact form:
\[p ( x ) = a [ (( x + \frac{b}{2 a} ))^{ 2} - \frac{b^{2}}{4 a^{2}} ] + c\]Distributing $a$ and combining the constant terms yields:
\[p ( x ) = a (( x + \frac{b}{2 a} ))^{ 2} - \frac{b^{2}}{4 a} + c\]which is the vertex form with:
\[h = \frac{b}{2 a} k = c - \frac{b^{2}}{4 a}\]Geometric interpretation
The algebraic identity underlying the method of completing the square allows for a direct geometric interpretation. Consider the following expression:
\[x^{2} + 6 x + 9\]Each term represents the area of a specific geometric region: $x^{2}$ corresponds to a square with side length $x$; $6 x$ represents the combined area of two rectangles, each measuring $x \times 3$; and $9$ denotes the area of a square with side length $3$. When these three regions are arranged around a common vertex, they tile a larger square with side length $x + 3$, thereby confirming the identity:
\[x^{2} + 6 x + 9 = ( x + 3 )^{2}\]
This geometric reasoning applies precisely when the constant term equals the square of half the linear coefficient, as in a polynomial with a repeated root. Setting the expression equal to zero yields the equation $( x + 3 )^{2} = 0$, whose unique solution is $x = - 3$. In the general case, completing the square provides the necessary correction algebraically, even when the resulting configuration does not correspond to a concrete geometric realisation over the positive real numbers.
When the coefficients are small integers and the polynomial factors readily, this geometric approach is often more straightforward than using the quadratic formula. Its effectiveness decreases when the leading coefficient or the linear term contains fractions or irrational numbers, as the arithmetic becomes more complex and the quadratic formula is generally preferable.
Example 1
An application of the method can be demonstrated using the following quadratic equation:
\[3 x^{2} - 4 x - 1 = 0\]The constant term is moved to the right-hand side, and both sides are divided by $3$:
\[x^{2} - \frac{4}{3} x = \frac{1}{3}\]The value added to both sides is the square of half the coefficient of $x$, specifically:
\[(( \frac{1}{2} \cdot \frac{4}{3} ))^{ 2} = (( \frac{2}{3} ))^{ 2} = \frac{4}{9}\] \[x^{2} - \frac{4}{3} x + \frac{4}{9} = \frac{1}{3} + \frac{4}{9}\]At this stage, the left-hand side forms a perfect square trinomial:
\[(( x - \frac{2}{3} ))^{ 2} = \frac{3}{9} + \frac{4}{9} = \frac{7}{9}\]Taking the square root of both sides yields
\[x - \frac{2}{3} = \pm \frac{\sqrt{7}}{3}\]The equation has two real roots:
\[x = \frac{2 \pm \sqrt{7}}{3}\]When the coefficients are not small integers, completing the square is typically more laborious than directly applying the quadratic formula. The latter method is generally preferable in such cases.
Derivation of the quadratic formula
A principal application of completing the square is that it produces the quadratic formula as a direct consequence, rather than as an independent result. The derivation begins with the general quadratic equation:
\[a x^{2} + b x + c = 0 a \neq 0.\]Dividing both sides by $a$ and transposing the constant term to the right-hand side:
\[x^{2} + \frac{b}{a} x = - \frac{c}{a}\]Adding $(( \frac{b}{2 a} ))^{ 2}$ to both sides completes the square on the left:
\[x^{2} + \frac{b}{a} x + (( \frac{b}{2 a} ))^{ 2} = (( \frac{b}{2 a} ))^{ 2} - \frac{c}{a}\]This yields:
\[(( x + \frac{b}{2 a} ))^{ 2} = \frac{b^{2} - 4 a c}{4 a^{2}}\]Taking the square root of both sides and solving for $x$ we obtain:
\[x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}\]This derivation shows that the quadratic formula is a direct consequence of completing the square applied to the general quadratic equation.
The expression $b^{2} - 4 a c$, which appears under the square root in the quadratic formula, is called the discriminant of the equation. Its sign determines the nature of the roots:
- When $b^{2} - 4 a c > 0$ the equation has two distinct real roots.
- When $b^{2} - 4 a c = 0$ it has a single repeated real root.
- When $b^{2} - 4 a c < 0$ there are no real roots, since the square root of a negative number is not defined in $\mathbb{R}$.