Complete the Square for 2𝑥^2−6𝑥+7

Introduction

Many students lose marks because they nearly complete the square, but miss one small adjustment. In this post, we’ll learn how to complete the square cleanly and present the final answer exactly in the required form \(a(x+b)^2 + c\). This is a high-frequency skill for O Level A Maths — especially in quadratic manipulation questions.

The Question / Scenario Explanation

Question (as shown): Express \(2x^2 – 6x + 7\) in the form \(a(x+b)^2 + c\), where \(a\), \(b\) and \(c\) are constants.

Step-by-Step Solution / Explanation

Step 1: Factor out the coefficient of \(x^2\)

We start by taking \(2\) outside the first two terms:
\(2x^2 – 6x + 7 = 2(x^2 – 3x) + 7\)

Step 2: Complete the square inside the bracket

To complete the square, take half of \(-3\), which is \(-\tfrac{3}{2}\), then square it:
\((- \tfrac{3}{2})^2 = \tfrac{9}{4}\)

Add and subtract \(\tfrac{9}{4}\) inside the bracket:
\(2(x^2 – 3x + \tfrac{9}{4} – \tfrac{9}{4}) + 7\)

Group the perfect square:
\(2((x – \tfrac{3}{2})^2 – \tfrac{9}{4}) + 7\)

Step 3: Simplify the constants

Distribute the \(2\):
\(2(x – \tfrac{3}{2})^2 – 2\cdot\tfrac{9}{4} + 7\)

Compute the constant term:
\(2\cdot\tfrac{9}{4} = \tfrac{18}{4} = \tfrac{9}{2}\)

So we have:
\(2(x – \tfrac{3}{2})^2 – \tfrac{9}{2} + 7\)

Convert \(7\) into halves: \(7 = \tfrac{14}{2}\). Combine constants:
\(-\tfrac{9}{2} + \tfrac{14}{2} = \tfrac{5}{2}\)

✅ Final Answer:
\(2(x – \tfrac{3}{2})^2 + \tfrac{5}{2}\)

Key Concepts Students Must Know

• To complete the square, use the pattern: \(x^2 + px = (x + \tfrac{p}{2})^2 – (\tfrac{p}{2})^2\).
• Always factor out the coefficient of \(x^2\) first if it’s not \(1\).
• Keep fractions neat — they are expected in O Level A Maths.

Exam Tips / Common Mistakes

Exam Tips

• When you complete the square, do it in three clean lines: factor → add/subtract → simplify.
• If the coefficient outside is \(2\), remember it multiplies BOTH the square term and the constant adjustment.
• Final form must match \(a(x+b)^2 + c\) exactly.

Common Mistakes

• Forgetting to factor out \(2\) first (this changes the square step).
• Adding \(\tfrac{9}{4}\) but forgetting to subtract \(\tfrac{9}{4}\) (must keep expression equivalent).
• Distributing \(2\) incorrectly: \(2((x-\tfrac{3}{2})^2 – \tfrac{9}{4})\) becomes \(2(x-\tfrac{3}{2})^2 – \tfrac{9}{2}\), not \(-\tfrac{9}{4}\).

Parent Insight

Completing the square is not just a “one question skill” — it supports many O Level topics like quadratic graphs, solving quadratics, and optimisation-style problems. When your child can complete the square confidently, they become faster and more accurate in Paper 1 questions that test algebraic manipulation.

Conclusion

This question is a classic example of how to complete the square correctly: factor out \(2\), form the perfect square \((x – \tfrac{3}{2})^2\), then simplify the constant term to get \(\tfrac{5}{2}\). Final form: \(2(x – \tfrac{3}{2})^2 + \tfrac{5}{2}\).

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