Question

what is the value of the discriminant, ( b^{2}-4ac ), for the quadratic equation ( 0=-2x^{2}-3x + 8 ), and what does it mean about the number of real solutions the equation has?
the discriminant is -55, so the equation has 2 real solutions.
the discriminant is -55, so the equation has no real solutions.
the discriminant is 73, so the equation has 2 real solutions.
the discriminant is 73, so the equation has no real solutions.

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here equation is \(0=-2x^2 - 3x + 8\) or \(-2x^2 - 3x + 8 = 0\). So \(a=-2\), \(b=-3\), \(c = 8\).

Step2: Calculate discriminant

Discriminant \(D=b^2-4ac\). Substitute values: \(D=(-3)^2-4\times(-2)\times8\).
First, \((-3)^2 = 9\), \(4\times(-2)\times8=-64\), so \(D = 9-(-64)=9 + 64 = 73\).

Step3: Analyze number of real solutions

If discriminant \(D>0\), quadratic has 2 real solutions; \(D = 0\), 1 real solution; \(D<0\), no real solutions. Here \(D = 73>0\), so 2 real solutions.

Answer:

The discriminant is 73, so the equation has 2 real solutions. (Option: The discriminant is 73, so the equation has 2 real solutions.)