J's answer to Lol's Secondary 2 Maths Singapore question.
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Recall the algebraic identity (a + b)² = a² + 2ab + b².
So what we need to do is to rewrite ¼ + x + x² into this form.
¼ + x + x²
= (½)² + 2(½)(x) + x²
= (½ + x)²
Here, our a = ½ and b = x
Alternatively, since a and b are actually interchangeable since a + b = b + a.
So we could rewrite it as (x + ½)²
= x² + 2(x)(½) + (½)²
= x² + x + ¼
Here, our a = x and b = ½
So what we need to do is to rewrite ¼ + x + x² into this form.
¼ + x + x²
= (½)² + 2(½)(x) + x²
= (½ + x)²
Here, our a = ½ and b = x
Alternatively, since a and b are actually interchangeable since a + b = b + a.
So we could rewrite it as (x + ½)²
= x² + 2(x)(½) + (½)²
= x² + x + ¼
Here, our a = x and b = ½
Date Posted:
4 years ago
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Or, I could factor out a 1/4 to make it easier to the eye.
1/4 + x + x^2
= 1/4 (1 + 4x + 4x^2)
= ...
And then we split 1/4 as 1/2 x 1/2 before integrating them into each bracket.
Students should build intuition in knowing how to spot factors out from expressions.
This example is fairly simple anyway.
Things like 1/9 = (⅓)² , ¼ = (½)² , x = ½(2)x and ¼x = 2(⅛)x = 2(½)(¼x) = 2(¼)(½x)
At higher levels, these would become second nature