To solve the equation x(4x^2+3)/(2x+1)^3 = 7, we can follow these steps:
1. Simplify the denominator by expanding (2x+1)^3 using the binomial formula:
(2x+1)^3 = 8x^3 + 12x^2 + 6x + 1
2. Substitute this expression and simplify the numerator:
x(4x^2+3)/(8x^3+12x^2+6x+1) = 7
4x^3 + 3x = 56x^3 + 84x^2 + 42x + 7
52x^3 + 84x^2 + 39x - 7 = 0
3. Factor the polynomial. Since there are no obvious common factors, we can use numerical methods or the rational root theorem to find a root. One possible root is x = 1/2:
52(1/2)^3 + 84(1/2)^2 + 39(1/2) - 7 = 0
13 + 21 + 39/2 - 7 = 0
So x = 1/2 is a root of the polynomial.
4. Divide the polynomial by x - 1/2 to obtain a quadratic factor. We can use polynomial division or synthetic division to do this. The result is:
52x^2 + 105x + 14 = 0
5. Solve the quadratic equation using the quadratic formula or factoring. The solutions are:
x = (-105 ± sqrt(105^2 - 4(52)(14))) / (2(52))
x = (-105 ± sqrt(11041)) / 104
x = (-105 ± 105) / 104 or x = (-105 ± sqrt(11041)) / 104
x = -1 or x ≈ 0.1417 or x ≈ -1.6276
Therefore, the solutions to the equation x(4x^2+3)/(2x+1)^3 = 7 are x = -1, x ≈ 0.1417, and x ≈ -1.6276.
Comments
Post a Comment