Determine the number and type of solutions for the equation \( 5 x^{2}-18 x-8=0 \) Two different rational-number solutions One repeated irrational-number solution Two different irrational-number solutions One repeated rational-number solution Two imaginary-number solutions

To determine the number and type of solutions for the equation \( 5x^{2}-18x-8=0 \), we can use the discriminant method. The discriminant \( D \) is given by the formula \( D = b^2 - 4ac \), where \( a = 5 \), \( b = -18 \), and \( c = -8 \). Calculating the discriminant: \[ D = (-18)^2 - 4(5)(-8) = 324 + 160 = 484. \] Since \( D > 0 \), this means there are two distinct real solutions. Next, we check if they are rational or irrational by determining the square root of the discriminant: \[ \sqrt{D} = \sqrt{484} = 22, \] which is a rational number. Therefore, the solutions are rational. Thus, the type of solutions for the equation \( 5x^{2}-18x-8=0 \) is **two different rational-number solutions**. Here's some extra thought on context and applications: But did you know? Rational solutions often mean the equation can be factored over the integers! If you ever find yourself in a crunch during algebra, this is a great sign that your equation may simplify nicely! And if you want to dive even deeper into quadratic equations, check out the quadratic formula derivation! It's a fantastic journey through algebra that not only teaches methodology but also reinforces the relationship between coefficients and roots. Give it a read and you’ll unveil more mathematical treasures!