Worked example 1
Complete a non-unitary square
Write 2x² − 8x + 3 in completed-square form
- Factor 2 from the x-terms: 2(x² − 4x) + 3
- Use x² − 4x = (x − 2)² − 4.
- Simplify the constant.
Answer: 2(x − 2)² − 5
Higher Mathematics
Higher Maths Completing the square and quadratic inequalities
Practise completing the square and quadratic inequalities for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Complete the square for unitary and non-unitary quadratics, then solve quadratic inequalities.
Before you start
- Factorise simple quadratics.
- Solve quadratic equations.
- Interpret where a graph is above or below the x-axis.
Explanation
Completing the square rewrites a quadratic to reveal its turning point and range. For a non-unitary quadratic, factor the leading coefficient from the x-terms first.
To solve a quadratic inequality, find the boundary roots and use the parabola's sign on each interval.
Method and rules
- ax² + bx + c = a(x + )² + c −
- For an upward parabola, f(x) > 0 outside two distinct roots.
- Use open endpoints for < or > and closed endpoints for ≤ or ≥.
Worked examples
Worked example 2
Method check
Solve (x + 2)(x − 3) > 0.
- The upward quadratic is positive outside its roots.
So: x < −2 or x > 3
Watch out
- Completing the square before accounting for the leading coefficient.
- Giving only roots instead of inequality intervals.
- Including a boundary when the inequality is strict.
Exam reminder
For completing the square and quadratic inequalities, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.