a) \( x=-5 \) and \( x=-1 \) b) \( x=7 \) and \( x=1 \) c) \( x=5 \) and \( x=1 \) d) \( x=2 \) and \( x=-2 \) Use the quadratic formula to solve the equations, \[ =5+\sqrt{3} z^{-7 x^{2}-5 x+1=0} \] 38. \( x^{2}-4 x-12=0 \)

To solve the quadratic equation \( x^{2}-4x-12=0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a=1 \), \( b=-4 \), and \( c=-12 \), we first calculate the discriminant: \[ b^2 - 4ac = (-4)^2 - 4(1)(-12) = 16 + 48 = 64. \] Now, plugging into the formula gives us: \[ x = \frac{4 \pm \sqrt{64}}{2} = \frac{4 \pm 8}{2}. \] Thus, the solutions are \( x = 6 \) and \( x = -2 \). Ever thought how math can time-travel? Quadratic equations have been around since ancient civilizations, such as the Babylonians, who had their methods to solve them long before the quadratic formula was formalized in the 17th century! It's a beautiful blend of art and logic. In real life, quadratic equations pop up more than you'd think! From calculating the trajectory of thrown objects in sports to determining areas in architecture, understanding these equations can even help you optimize daily scenarios like budgeting or maximizing your garden space. So, don’t underestimate those pesky equations!