Simplify the radical equation. State the non-permissive values.
10๐ฅ^2+๐ฅโ2/2๐ฅ^2โ9๐ฅโ18/ โ4๐ฅโ6/4๐ฅ+6=4๐ฅ/๐ฅโ6
To simplify the given radical equation, we first need to rationalize the denominator.
The given equation is:
((10x^2 + x - 2) / (2x^2 - 9x - 18)) / ((-4x - 6) / (4x + 6)) = 4x / (x - 6)
In rationalization, we multiply both the numerator and denominator of the whole fraction by the conjugate of the denominator in the other fraction.
Using that, we multiply both the numerator and denominator of the large fraction by (4x + 6)/(4x + 6):
((10x^2 + x - 2) / (2x^2 - 9x - 18)) * ((4x + 6) / (4x + 6)) / ((-4x - 6) / (4x + 6)) = (4x / (x - 6)) * ((4x + 6) / (4x + 6))
Simplifying this expression:
((10x^2 + x - 2) * (4x + 6)) / ((2x^2 - 9x - 18) * (-4x - 6)) = (4x * (4x + 6)) / ((x - 6) * (4x + 6))
Simplifying the numerators and denominators individually:
(40x^3 + 20x^2 + 60x^2 + 30x - 80x - 120) / ((2x + 3)(x - 2)(-2x - 3)(x - 6)) = (16x^2 + 24x) / ((x - 6)(4x + 6))
Combining like terms:
(40x^3 + 80x^2 - 50x - 120) / ((2x + 3)(x - 2)(-2x - 3)(x - 6)) = (16x^2 + 24x) / ((x - 6)(4x + 6))
Now, we can cancel out common factors:
(5x + 6)(8x^2 - 4) / ((2x + 3)(-2x - 3)(x - 6)) = 8x(2x + 3) / ((x - 6)(4x + 6))
Simplifying further, we get:
[(5x + 6)(2x - 2)(4x + 6)] / [-(2x + 3)(2x + 3)(x - 6)] = [8x(2x + 3)] / [2(x - 6)(2x + 3)]
The simplified radical equation is:
(5x + 6)(2x - 2)(4x + 6) / [-(2x + 3)(2x + 3)(x - 6)] = [8x(2x + 3)] / [2(x - 6)(2x + 3)]
The non-permissive values are x = -3/2, -2, 6.