Completing the square

Study Notes

Completing the square is a method used to solve quadratic equations by rewriting them in the form $p(x+q)^2 + r = 0$, where $p$, $q$, and $r$ are integers. This method simplifies the equation to have $x$ appearing only once.

Exam Tips

Key Definitions to Remember

• Completing the Square — A method to solve quadratic equations by rewriting them in the form $p(x+q)^2 + r = 0$.

Common Confusions

• Forgetting to divide the entire equation by the coefficient of $x^2$ before completing the square.
• Incorrectly balancing the equation after adding and subtracting the square of half the coefficient of $x$.

Typical Exam Questions

• How do you express $2x^2 + 4x - 14 = 0$ in the form $p(x+q)^2 + r = 0$? Divide by 2, complete the square to get $(x+1)^2 - 8 = 0$.
• Solve $x^2 - 8x + 11 = 0$ by completing the square. $(x-4)^2 - 5 = 0$, so $x = 4 \pm \sqrt{5}$.
• Solve $7x^2 = 4(1 + x)$ by completing the square. Rearrange and complete the square to find $x$.

What Examiners Usually Test

• Ability to rewrite quadratic equations in completed square form.
• Solving quadratic equations using the completed square method.
• Providing answers in surd form for exact solutions.