Equations and Inequalities

Linear and quadratic equations by every method the syllabus allows, simultaneous equations including the linear-and-quadratic case, and inequalities on a number line.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ครอบคลุม ม.3 บทที่ 1 อสมการ · บทที่ 3 สมการกำลังสอง · บทที่ 6 ระบบสมการ
Common Core (US): A-REI, 8.EE
Cambridge: IGCSE 0580 — Topic 2: Algebra and graphs (Core and Extended)

3.1 Three ways to solve a quadratic

Method	Use it when	Tier
factorising	it factorises with whole numbers	Core and Extended
the formula	it does not factorise, or you cannot see how	Extended
completing the square	you also need the turning point	Extended

Choosing a method

The quadratic formula

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

For $a x^{2} + b x + c = 0$ . The discriminant $b^{2} - 4 a c$ tells you how many real roots there are before you find any.

ตัวอย่าง 1

Solve $2 x^{2} - 7 x + 3 = 0$ by factorising.

วิธีทำ

1. $a c = 6$ ; two numbers multiplying to $6$ and adding to $- 7$ : $- 1$ and $- 6$
2. $2 x^{2} - x - 6 x + 3 = x (2 x - 1) - 3 (2 x - 1)$
3. $(2 x - 1) (x - 3) = 0$

ตอบ $x = \frac{1}{2}$ or $x = 3$

ตัวอย่าง 2

Solve $3 x^{2} + 5 x - 1 = 0$ , giving answers to 2 decimal places.

วิธีทำ

1. $a = 3$ , $b = 5$ , $c = - 1$
2. $b^{2} - 4 a c = 25 + 12 = 37$
3. $x = \frac{- 5 \pm 37}{6}$

ตอบ $x = 0.18$ or $x = - 1.85$

ข้อควรระวัง

Substitute $b$ WITH its sign. With $b = - 7$ , the formula needs $- b = + 7$ ; writing $- 7$ there is the error that produces two wrong roots from correct working.

ทำไมถึงเป็นแบบนั้น

Completing the square is worth the effort even when the formula would do, because $(x - h)^{2} + k$ hands you the turning point $(h, k)$ for free — and the graph questions in the next chapter ask for exactly that.

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