Calculate the discrininat 2. \( x^{2}-64=0 \)

To find the discriminant of the equation \( x^{2}-64=0 \), first rewrite it in standard form: \( x^{2} + 0x - 64 = 0 \). The discriminant \( D \) is given by the formula \( D = b^{2} - 4ac \). Here, \( a = 1 \), \( b = 0 \), and \( c = -64 \). Plugging in these values, we get: \[ D = 0^{2} - 4(1)(-64) = 0 + 256 = 256. \] So, the discriminant is \( 256 \). Now, a discriminant serves as a key to understanding the nature of the roots. If \( D > 0 \), as it is here, it means there are two distinct real roots. Imagine two friends escaping a room – they have different paths to success, and here they certainly do, with the potential to uncover two unique solutions for \( x \)! Also, once you know the discriminant, you can quickly assess the equation's root dynamics. For \( x^{2}-64=0 \), we can deduce that since \( D = 256 \), your next step might be to find those roots using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \). How exciting to see where those paths lead!