ACT Preparation / Unit 11 ACT 1017 worked example, 60 to 90 seconds
Completing the Square to Vertex Form
The idea
To rewrite ax² + bx + c in vertex form, work in four steps. First, factor a out of the x² and x terms. Second, inside the parentheses, add and subtract the square of half the x-coefficient. Third, move the subtracted amount outside, multiplying it by a. Fourth, write the perfect square, combine the constants, and read the vertex.
In practice
Worked example: f(x) = 2x² − 12x + 7. Step one: 2(x² − 6x) + 7. Step two: half of −6 is −3, and its square is 9: 2(x² − 6x + 9 − 9) + 7. Step three: 2(x² − 6x + 9) − 18 + 7. Step four: 2(x − 3)² − 11. The vertex is (3, −11), a minimum.
Where people slip
The amount moved outside is multiplied by a. Subtracting 9 instead of 18 here gives a vertex of (3, −2), which is wrong.
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