Expressions and Formulae

Expanding double brackets, factorising them back, and rearranging a formula so it answers the question you actually have.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.2 บทที่ 6 พหุนาม และบทที่ 11 การแยกตัวประกอบ
Common Core (US): A-SSE, A-APR, 7.EE
Cambridge: Lower Secondary Stage 8 — Algebra: expressions, equations and formulae

4.1 Expanding brackets

Single and double brackets

a (b + c) = ab + a c (a + b) (c + d) = a c + a d + b c + b d

Every term in the first bracket meets every term in the second — four products from two pairs, and none of them optional.

ตัวอย่าง

Expand and simplify $(x + 3) (x - 5)$

วิธีทำ

1. $x \times x = x^{2}$
2. $x \times (- 5) = - 5 x$
3. $3 \times x = 3 x$
4. $3 \times (- 5) = - 15$
5.Collect: $- 5 x + 3 x = - 2 x$

ตอบ $x^{2} - 2 x - 15$

Three worth recognising instantly

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

These come back in the next section read backwards, and again in Stage 9 for solving quadratics.

ข้อควรระวัง

$(a + b)^{2}$ is not $a^{2} + b^{2}$ . Put $a = 3$ and $b = 4$ : the left is $49$ , the right is $25$ . The missing $2 ab$ is the middle term, and forgetting it is the most common algebra error at this stage.

เคล็ดลับ

Check any expansion by substituting a number — $x = 2$ , say — into both the original and your answer. If they disagree, the expansion is wrong, and it takes five seconds to find out.

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