To find the solutions for this cubic equation, we can use the rational root theorem to try possible rational roots. The rational root theorem states that if a polynomial has a rational root p/q, then p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the constant term is 3 and the leading coefficient is 8. The possible factors of 3 are ±1, ±3 and the possible factors of 8 are ±1, ±2, ±4, ±8. So, the possible rational roots are ±1, ±1/2, ±3, ±3/2.
By trying these possible roots, we find that x = 1/2 is a solution to the equation. Dividing the equation by (x - 1/2) gives:
[tex]8 {x}^{2} + 12x - 6 = 0[/tex]
This is a quadratic equation that can be solved using the quadratic formula:
The solutions are approximately: x ≈ -1.3542 x ≈ 0.2292
Therefore, the solutions to the cubic equation are: x = 1/2 x ≈ -1.3542 x ≈ 0.2292
To find the solutions for this cubic equation, we can use the rational root theorem to try possible rational roots. The rational root theorem states that if a polynomial has a rational root p/q, then p is a factor of the constant term and q is a factor of the leading coefficient.
In this case, the constant term is 3 and the leading coefficient is 8. The possible factors of 3 are ±1, ±3 and the possible factors of 8 are ±1, ±2, ±4, ±8. So, the possible rational roots are ±1, ±1/2, ±3, ±3/2.
By trying these possible roots, we find that x = 1/2 is a solution to the equation. Dividing the equation by (x - 1/2) gives:
[tex]8 {x}^{2} + 12x - 6 = 0[/tex]
This is a quadratic equation that can be solved using the quadratic formula:
The solutions are approximately:
x ≈ -1.3542
x ≈ 0.2292
Therefore, the solutions to the cubic equation are:
x = 1/2
x ≈ -1.3542
x ≈ 0.2292