AQA · Mathematics · 8300
GCSE Maths: Quadratics: solving (completing the square)
Algebra · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths textbook exercise and practice questions for Quadratics: solving (completing the square) (Corbettmaths Video 267), and every card links back to the exact page it came from. Practice tests generate fresh variations of these questions, so the numbers change every attempt. Source material © Corbettmaths — used with attribution.
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What is the general form of a quadratic expression when completed to the square?
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(x + a)² + b, where a and b are constants.
If x² + px + q is rewritten by completing the square as (x − 5)² + 31, what is the value of p?
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p = −10. Expanding (x − 5)² + 31 gives x² − 10x + 25 + 31 = x² − 10x + 56.
If x² + px + q is rewritten by completing the square as (x − 5)² + 31, what is the value of q?
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q = 56. Expanding (x − 5)² + 31 gives x² − 10x + 25 + 31 = x² − 10x + 56.
If x² − 4x + b ≡ (x + a)² + 11, what is the value of a?
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a = −2. The coefficient of x is −4, so half of it is −2.
Why does the equation x² + 4x + 15 = 0 have no real solutions?
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When completed to the square, it becomes (x + 2)² + 11 = 0, giving (x + 2)² = −11. Since a square cannot equal a negative number (in real numbers), there are no real solutions.
What is the minimum point of the curve y = x² − 6x + 1?
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(3, −8). Completing the square gives y = (x − 3)² − 8, so the minimum occurs at x = 3, y = −8.
How can completing the square be used to find the minimum point of a parabola y = x² + bx + c?
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Rewrite as y = (x + a)² + k. The minimum point is at (−a, k), since the squared term is minimized when it equals zero.
If the curve y = x² − 10x + 20 is written as (x − a)² − b, what is the equation of its line of symmetry?
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x = 5. The line of symmetry passes through the x-coordinate of the vertex, which is x = a when in the form (x − a)² − b.
How do you express 3x² + 18x − 1 in completed square form?
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Factor out the coefficient of x² first: 3(x² + 6x) − 1, then complete the square inside: 3((x + 3)² − 9) − 1 = 3(x + 3)² − 28.
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