Chapter 2

Quadratic Functions and Inequalities

Completing the square as a tool rather than a trick, the discriminant as a question about how many times a line meets a curve, and quadratic inequalities solved without guessing.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ต่อยอดจาก ม.3 สมการกำลังสอง และ ม.4 อสมการ — 0606 เพิ่มการใช้ดิสคริมิแนนต์กับเส้นตรงที่ตัดเส้นโค้ง
Common Core (US): A-SSE, A-REI, F-IF
Cambridge: IGCSE 0606 — Topic 2: Quadratic functions

2.1 Completed-square form

The form, and what each part tells you

y = a (x - h)^{2} + k vertex (h, k)

If $a > 0$ the vertex is a minimum; if $a < 0$ it is a maximum. The line $x = h$ is the axis of symmetry.

Completing the square when the leading coefficient is not 1

1Take the coefficient of $x^{2}$ out of the FIRST TWO terms only
2Halve the number now multiplying $x$ inside the bracket, and square it
3Add and subtract that square inside the bracket
4Multiply the subtracted part back out and combine it with the constant

ตัวอย่าง

Write $2 x^{2} - 12 x + 23$ in the form $a (x - h)^{2} + k$ .

วิธีทำ

1. $2 (x^{2} - 6 x) + 23$
2.Half of $- 6$ is $- 3$ , and $(- 3)^{2} = 9$
3. $2 [(x - 3)^{2} - 9] + 23$
4. $2 (x - 3)^{2} - 18 + 23$

ตอบ $2 (x - 3)^{2} + 5$ — a minimum of $5$ at $x = 3$

เคล็ดลับ

Completed-square form answers three different exam questions at once: the vertex, the range, and the least or greatest value. Producing it early often collapses three parts of a question into one calculation.

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