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Completing the Square
Chapter summary, hard words and model exam answers for ICSE Class 10 Hindi.
Free online summary and notes (ICSE Class 10 Hindi). Read it here, no PDF download needed.
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Mathematics · CBSE 10 · ICSE 10 · GCSE (AQA, Edexcel, OCR)
Summary
Many quadratics will not factorise into neat brackets. Completing the square always works, and it also shows the turning point of the parabola, which factorising never does.
Because (x + b/2)² = x² + b x + (b/2)², the extra (b/2)² must be subtracted again so the value is unchanged. That single idea is the whole method.
Write a(x² + (b/a)x) + c, complete the square in the bracket, then multiply a back in. Watch signs carefully when a is negative.
If a > 0 the curve opens up and k is the minimum; if a < 0 it opens down and k is the maximum. Note (x + 3)² means h = −3, a common sign slip.
Hard words & meanings
| completing the square | rewriting a x² + b x + c as a(x − h)² + k by adding and subtracting (b/2)² |
| quadratic | an expression with highest power 2, of the form a x² + b x + c |
| coefficient | the number multiplying a term, e.g. b is the coefficient of x |
| vertex form | the form a(x − h)² + k from which the turning point is read directly |
| vertex | the turning point of a parabola, at (h, k) in vertex form |
| axis of symmetry | the vertical line x = h that mirrors the parabola |
| parabola | the U-shaped graph of a quadratic function |
| discriminant | b² − 4ac, the part under the root sign that decides how many solutions exist |
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