Respuesta :

According to the discriminant of the quadratic equation, it will have equal roots for p = -16.

What is the discriminant of a quadratic equation and how does it influence the solutions?

A quadratic equation is modeled by:

[tex]y = ax^2 + bx + c[/tex]

The discriminant is:

[tex]\Delta = b^2 - 4ac[/tex]

The solutions are as follows:

  • If [tex]\mathbf{\Delta > 0}[/tex], and it is a perfect square, it has 2 rational solutions.
  • If [tex]\mathbf{\Delta > 0}[/tex], and it is not a perfect square, it has 2 irrational solutions.
  • If [tex]\mathbf{\Delta = 0}[/tex], it has equal roots.
  • If [tex]\mathbf{\Delta < 0}[/tex], it has 2 complex solutions.

In this problem, the equation is:

[tex]f(x) = -x^2 + 8x + p[/tex]

Hence the coefficients are a = -1, b = 8, c = p.

It has equal roots if [tex]\Delta = 0[/tex], hence:

[tex]b^2 - 4ac = 0[/tex]

[tex]8^2 + 4p = 0[/tex]

[tex]4p = -64[/tex]

[tex]p = -16[/tex]

More can be learned about the discriminant of a quadratic equation at https://brainly.com/question/19776811

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