Mathematics

Mathematics

Problem

Let E(x)=(x2+7)2−49E(x)=(x^{2}+7)^{2}-49. a. (i) Show that E(x)E(x) can be expressed as the difference of two perfect squares. b. (ii) Factorise E(x)E(x) completely, writing your answer as a product of linear or irreducible quadratic factors. c. (iii) Hence find all real roots of the equation (x2+7)2−49=0(x^{2}+7)^{2}-49=0. Present your answer in simplest form.
Intuitive explanation

The key insight is recognizing that E(x)E(x) is already almost in the form a2−b2a^2 - b^2—you just need to see what the two squares are. Once you identify the difference of squares pattern, you can apply the factorization a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b) to break it down, then look for further factorization opportunities. From there, setting each factor to zero reveals which real solutions actually exist.

Intuitive explanation

The key insight is recognizing that E(x)E(x) is already almost in the form a2−b2a^2 - b^2—you just need to see what the two squares are. Once you identify the difference of squares pattern, you can apply the factorization a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b) to break it down, then look for further factorization opportunities. From there, setting each factor to zero reveals which real solutions actually exist.

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