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Solve the equation [tex]28x^3 + 88x^2 - 37x - 7 = 0[/tex] given that [tex]-\frac{7}{2}[/tex] is a zero of [tex]f(x) = 28x^3 + 88x^2 - 37x - 7[/tex].

Answer :

Final Answer:

The solution to the equation [tex]28x^3 + 88x^2 - 37x - 7 = 0[/tex] given that -7/2 is a zero of [tex]f(x) = 28x^3 + 88x^2 - 37x - 7[/tex] is x = -1/2.

Explanation:

To solve the equation [tex]28x^3 + 88x^2 - 37x - 7 = 0[/tex], we first apply the factor theorem:

[tex]f(-7/2) = 28(-7/2)^3 + 88(-7/2)^2 - 37(-7/2) - 7 = 0.[/tex]

This confirms that -7/2 is a zero of f(x), so x + 7/2 is a factor of f(x).

Now, divide f(x) by x + 7/2:

[tex](28x^3 + 88x^2 - 37x - 7) / (x + 7/2) = (x + 7/2)(28x^2 + 14x - 1) / (x + 7/2) = 28x^2 + 14x - 1.[/tex]

Apply the quadratic formula to solve [tex]28x^2 + 14x - 1 = 0:[/tex]

x = (-b ± √([tex]b^2[/tex] - 4ac)) / (2a).

x = (-14 ± √([tex]14^2[/tex] - 4(28)(-1))) / (2(28)).

x = (-14 ± √(196 + 112)) / 56.

x = (-14 ± √308) / 56.

x = (-14 ± 2√77) / 56.

x = (-7 ± √77) / 28.

Thus, the roots of the given cubic equation are -7/2, -1/2, and 1/2, and the solution to [tex]28x^3 + 88x^2 - 37x - 7 = 0[/tex] given that -7/2 is a zero of [tex]f(x) = 28x^3 + 88x^2 - 37x - 7 is x = -1/2[/tex].

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