Completing the Square (DP IB Analysis & Approaches (AA): HL): Revision Note

Completing the square

Why is completing the square for quadratics useful?

• Completing the square gives the maximum or minimum of a quadratic function

  • This can be used to define the range of the function

• It gives the vertex when drawing the graph

• It can be used to solve quadratic equations

• It can be used to derive the quadratic formula

How do I complete the square for a monic quadratic of the form x²+bx+c?

• Halve the value of $b$ and write down ${(x+\frac{b}{2})}^2$

  • This is because ${(x+\frac{b}{2})}^2=x^2+bx+{\left(\frac{b}{2}\right)}^2$

    • so it is similar to $x^2+bx+c$

• Subtract the unwanted ${\left(\frac{b}{2}\right)}^2$ term and add on the constant $c$

  • ${(x+\frac{b}{2})}^2-{\left(\frac{b}{2}\right)}^2+c$

• For example, $x^2-4x+5$

  • ${\left(x-2\right)}^2=x^2-4x+4$

  • $x^2-4x+5={\left(x-2\right)}^2-4+5={\left(x-2\right)}^2+1$

How do I complete the square for a non-monic quadratic of the form ax²+bx+c?

• Factorise out the $a$ from the terms involving $x$

  • $a(x^2+\frac{b}{a}x)+c$ 

  • Leaving the $c$ un-factorised will avoid working with lots of fractions

• Complete the square on the quadratic term inside the bracket

  • Halve $\frac{b}{a}$ and write ${(x+\frac{b}{2a})}^2$

    • This is because ${(x+\frac{b}{2a})}^2=x^2+\frac{b}{a}x+{\left(\frac{b}{2a}\right)}^2$

  • Subtract the unwanted ${\left(\frac{b}{2a}\right)}^2$ term

• Multiply through by $a$ and add on $c$

  • $a[{(x+\frac{b}{2a})}^2-{\left(\frac{b}{2a}\right)}^2]+c$

  • $a{(x+\frac{b}{2a})}^2-\frac{b^2}{4a}+c$

• For example, $2x^2+12x-3$

  • $2\left(x^2+6x\right)-3$

  • $2\left[{\left(x+3\right)}^2-9\right]-3$

  • $2{\left(x+3\right)}^2-18-3=2{\left(x+3\right)}^2-21$

How do I use completing the square to optimise with quadratics?

• You can use the completed square form to find the maximum or minimum of an expression

  • The optimal value of $a{\left(x-h\right)}^2+k$ is $k$

    • It is a minimum if $a$ is positive

    • It is a maximum if $a$ is negative

  • The optimal value occurs when $x=h$

• For example, suppose $P=0.5{\left(x-50\right)}^2+3000$ is the profit in dollars when $x$ units are sold

  • The maximum profit is $3000

  • This is achieved when 50 units are sold