How many solutions are in the solution set for each quadratic equation below?(3 points) 5x2=50 This quadratic equation would have solution(s). 3x2=−12 This quadratic equation would have solution(s). 8x2=0 This quadratic equation would have solution(s).

Answer 1

To determine the number of solutions for each quadratic equation, we can rearrange the equations into standard form (\(ax^2 + bx + c = 0\)) and analyze the discriminant (\(D = b^2 - 4ac\)).

1. For the equation \(5x^2 = 50\):

   • Rearrange it: \(5x^2 - 50 = 0\) or \(x^2 - 10 = 0\).
   • Here, \(a = 5\), \(b = 0\), \(c = -50\).
   • The discriminant: \(D = 0^2 - 4(5)(-50) = 0 + 1000 = 1000\) (which is greater than 0).
   • This equation has 2 real solutions.

2. For the equation \(3x^2 = -12\):

   • Rearrange it: \(3x^2 + 12 = 0\) or \(x^2 + 4 = 0\).
   • Here, \(a = 3\), \(b = 0\), \(c = 12\).
   • The discriminant: \(D = 0^2 - 4(3)(12) = 0 - 144 = -144\) (which is less than 0).
   • This equation has no real solutions (2 complex solutions).

3. For the equation \(8x^2 = 0\):

   • Rearrange it: \(8x^2 = 0\) or \(x^2 = 0\).
   • Here, \(a = 8\), \(b = 0\), \(c = 0\).
   • The discriminant: \(D = 0^2 - 4(8)(0) = 0\).
   • This equation has 1 real solution (a double root).

Summary:

1. \(5x^2 = 50\) has 2 solutions.
2. \(3x^2 = -12\) has no real solutions.
3. \(8x^2 = 0\) has 1 solution.