Determine the number and type of solutions for the equation \( -2 x^{2}+20 x-50=0 \) Two different irrational-number solutions Two imaginary-number solutions One repeated rational-number solution Two different rational-number solutions One repeated irrational-number solution Submit Question

To determine the number and type of solutions for the equation \( -2x^{2} + 20x - 50 = 0 \), we can use the discriminant method. First, we’ll rewrite the equation in standard quadratic form \( ax^2 + bx + c = 0 \). Here, \( a = -2 \), \( b = 20 \), and \( c = -50 \). Now, let's calculate the discriminant \( D = b^2 - 4ac \): \[ D = 20^2 - 4(-2)(-50) \] \[ D = 400 - 400 \] \[ D = 0 \] Since the discriminant is 0, this means there is one repeated rational-number solution. On a related note, did you know that the quadratic formula, \( x = \frac{-b \pm \sqrt{D}}{2a} \), is a magical tool that can unlock the mysteries of any quadratic equation? Whether your journey leads to rational, irrational, or even imaginary solutions, this little formula always has your back! If you're curious about the world of quadratics, you might find exploring Vieta's formulas intriguing! It beautifully relates the coefficients of a polynomial to sums and products of its roots, giving you another layer of understanding about how numbers dance together in equations.