Respuesta :
The given equation is the quadratic equation:
[tex]-x^2-6x-9=0[/tex]It is required to find the number and type of solutions.
To do this, the discriminant of the equation needs to be calculated.
Recall that the discriminant of a quadratic equation is:
[tex]b^2-4ac[/tex]Where a is the coefficient of x², b is the coefficient of x, and c is the constant.
Recall also that:
• If the discriminant equals zero, the equation has one repeated rational number solution.
,• If the discriminant is positive, the equation has two real solutions.
,• If the discriminant is negative, the equation has two imaginary solutions.
,• If the discriminant is a perfect square, then the equation has two real rational number solutions, it has two irrational number solutions.
Calculate the discriminant of the equation - x²- 6x - 9=0 by substituting a=-1, b=-6 and c=-9 into the discriminant formula:
[tex](-6)^2-4(-1)(-9)=36-36=0[/tex]Since the discriminant equals zero, it follows that the equation has one repeated rational number solution.
The equation has one repeated rational number solution.
Option D is correct.
