Solve simultaneous linear equations: (i) (x2 – x) (4×2 – 4x – 5) – 6

Solve simultaneous linear equations:

(i) (x² – x) (4x² – 4x – 5) – 6
(ii) x⁴ + 9x²y² + 81y⁴

Solution:

(i) (x² – x) (4x² – 4x – 5) – 6
(x² – x) [(4x² – 4x) – 5] – 6
(x² – x) [4(x² – x) – 5] – 6
Let us assume x² – x = q
So, q[4q – 5] – 6
4q² – 5q – 6
4q² – 8q + 3q – 6
4q(q – 2) + 3(q – 2)
(q – 2) (4q + 3)
Now, substitute the value of q
(x² – x – 2) (4(x² – x) + 3)
(x² – x – 2) (4x² – 4x + 3)
(x² – 2x + x – 2) (4x² – 4x + 3)
[x(x – 2) + 1(x – 2)] (4x² – 4x + 3)
(x – 2) (x + 1) (4x² – 4x + 3)
(ii) x⁴ + 9x²y² + 81y⁴
x⁴ + 9x²y² + 81y⁴
Above terms can be written as,
x⁴ + 18x²y² + 81y⁴ – 9x²y²
((x²)² + (2 × x² × 9y²) + (9y²)²) – 9x²y²
We know that, (a + b)² = a² + 2ab + b²
(x² + 9y²)² – (3xy)²
(x² + 9y² + 3xy) (x² + 9y² – 3xy)

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