Quadratic Expressions and Equations

Factorising a quadratic and using it to solve — the first equations in this book with two answers, and the property that makes finding them possible.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.2 บทที่ 11 การแยกตัวประกอบ และ ม.3 บทที่ 3 สมการกำลังสอง
Common Core (US): A-SSE, A-REI
Cambridge: Lower Secondary Stage 9 — Algebra: expressions, equations and formulae

5.1 Factorising quadratics

For $x^{2} + b x + c$

1Take out any common factor first — always the first move
2Find two numbers that MULTIPLY to $c$ and ADD to $b$
3Write $(x + p) (x + q)$ with those numbers
4Expand back to check

เคล็ดลับ

Start from the product, not the sum. There are only a handful of pairs multiplying to $c$ , and infinitely many adding to $b$ — searching the short list is the whole trick.

ตัวอย่าง 1

Factorise $x^{2} - 2 x - 15$

วิธีทำ

1. $c = - 15$ is negative, so the two numbers have different signs
2.Pairs multiplying to $- 15$ : $(1, - 15)$ , $(- 1, 15)$ , $(3, - 5)$ , $(- 3, 5)$
3. $3 + (- 5) = - 2$ , which is $b$

ตอบ $(x + 3) (x - 5)$

ตัวอย่าง 2

Factorise $2 x^{2} + 7 x + 3$

วิธีทำ

1. $a \neq = 1$ , so use $a c = 2 \times 3 = 6$
2.Two numbers multiplying to $6$ and adding to $7$ : $1$ and $6$
3.Split the middle: $2 x^{2} + x + 6 x + 3$
4.Group: $x (2 x + 1) + 3 (2 x + 1)$

ตอบ $(2 x + 1) (x + 3)$

ข้อควรระวัง

If grouping leaves two brackets that are NOT identical, the sign came out wrong when the second common factor was taken out. Go back and check whether it should have been negative.

Special forms to recognise instantly

a^{2} - b^{2} = (a + b) (a - b) a^{2} \pm 2 ab + b^{2} = (a \pm b)^{2}

$a^{2} + b^{2}$ does not factorise over the real numbers — there is nothing to find.

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