Question

Determine the roots of the equation `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)|` = 0

Answer 1

Now `|(1,4, 20),(1, -2, 5),(1, 2x, 5x^2)| = (5)|(1, 4, 4),(1, -2, 1),(1, 2x, x^2)|`

Taking 5 as a common factor from C₃

= `(5)(2)|(1, 2, 4),(1, -1, 1),(1, x, x^2)|`

Taking 2 as a common factor from C₂ 

= `10|(1, 2, 4),(1, -1, 1),(1, x, x^2)|`

= `10|(0, 3,3),(0, -1 - x, 1 - x^2),(1, x, x^2)|  {:("R"_1 -> "R"_1 - "R"_2),("R"_2 -> "R"_2 - "R"_3):}`

Expanding along C₁

`101[3(1 - x^2) - (- 1 - x)(3)]}`

= 10[3(1 – x²) + 3(1 + x)]

= 10[3(1 + x)(1 – x) + 3(1 + x)]

= 10[3(1 + x)(1 – x + 1)]

= 30(1 +x)(2 – x)

Given the determinant value is 0

⇒ 30(1 + x)(2 – x) = 0

⇒ 1 + x = 0 or 2 – x = 0

⇒ x = – 1 or x = 2

So, x = – 1 or 2.