All 303 families

RV-04-001 · Difference of squares with an error explanation

Consider 125x2−20125x^{2} - 20.

(a) Factorise this expression fully.

(b) A student says that the expression is equal to 5(5x−2)25(5x - 2)^2. Explain why the student is wrong.

Part (a) Factorising: DOTS (Difference of Two Squares) Booklet p. 2 & p. 6
Method
1) Take out HCF first
2) Use DOTS: a² − b² = (a + b)(a − b)
3) For part (b), expand (a − b)² with a 2×2 grid to show the non-zero middle x term
Worked Example (RV-04-001 · Part (a))
Factorise this expression fully.
Booklet p. 6 (DOTS): a2−b2=(a+b)(a−b)\mathbf{a^2 - b^2 = (a + b)(a - b)}
1. Take out HCF 5\mathbf{5}: 5(25x2−4)\quad 5(25x^2 - 4)
2. DOTS (square root each term): 25x2=5x\sqrt{25x^2} = 5x and 4=2\sqrt{4} = 2
⇒5(5x+2)(5x−2)\Rightarrow \mathbf{5(5x + 2)(5x - 2)}
1. Take out HCF (5)
2. DOTS: (a+b)(a−b)(a+b)(a-b)
5(5x+2)(5x−2)5(5x + 2)(5x - 2)
Part (b) Factorising: DOTS (Difference of Two Squares) Booklet p. 2 & p. 6
Method
1) Take out HCF first
2) Use DOTS: a² − b² = (a + b)(a − b)
3) For part (b), expand (a − b)² with a 2×2 grid to show the non-zero middle x term
Worked Example (RV-04-001 · Part (b))
A student says that the expression is equal to 5(5x−2)25(5x - 2)^2. Explain why the student is wrong.
Expand (5x−2)2(5x - 2)^2 using 2×2 Grid (Booklet p. 2):
×5x5x−2-2
5x5x25x225x^2−10x-10x
−2-2−10x-10x+4+4
5(5x−2)2=5(25x2−20x+4)=125x2−100x+205(5x - 2)^2 = 5(25x^2 \color{#dc2626}{- 20x} + 4) = 125x^2 \color{#dc2626}{- 100x} + 20
This has a non-zero middle xx term (−100x-100x), whereas 125x2−20125x^2 - 20 has no xx term!
Squaring the bracket creates a non-zero x term, whereas the given expression has no x term.