Transfer of Ideas Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
The idea of completing the square to solve x2+6x+5=0x^2+6x+5=0 transfers to converting x2+6x+5x^2+6x+5 to vertex form. Show both applications.

Solution

  1. 1
    Solving: x2+6x+5=0⇒(x+3)2−9+5=0⇒(x+3)2=4⇒x=−3±2=−1x^2+6x+5=0 \Rightarrow (x+3)^2-9+5=0 \Rightarrow (x+3)^2=4 \Rightarrow x=-3\pm 2 = -1 or −5-5.
  2. 2
    Vertex form: f(x)=x2+6x+5=(x+3)2−4f(x) = x^2+6x+5 = (x+3)^2-4. Vertex at (−3,−4)(-3,-4).
  3. 3
    The same algebraic manipulation (completing the square) solves the equation and reveals the geometric structure of the parabola.

Answer

x=−1 or x=−5;f(x)=(x+3)2−4 (vertex at (−3,−4))x=-1 \text{ or } x=-5;\quad f(x)=(x+3)^2-4 \text{ (vertex at }(-3,-4))
Transfer of ideas means recognising that a technique learned in one context (solving equations) applies in another context (graphing parabolas). The shared technique is completing the square.

About Transfer of Ideas

The ability to recognize that a technique or concept from one area of mathematics applies, possibly in adapted form, to a different area.

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More Transfer of Ideas Examples

Example 2 medium

The AM-GM inequality [formula] was originally about two positive numbers. Transfer the idea to prove

Example 3 easy

The factorisation [formula] transfers to factoring [formula]. Apply it.

Example 4 medium

The proof technique 'assume the hypothesis and derive the conclusion' (direct proof) from logic tran

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