Solve the quadratic equation \( 8 x^{2}+30 x+25=0 \). Show or explain how you arrived at your answer.

To solve the quadratic equation \( 8x^2 + 30x + 25 = 0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 8 \), \( b = 30 \), and \( c = 25 \). First, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 = 30^2 = 900 \] \[ 4ac = 4 \times 8 \times 25 = 800 \] \[ b^2 - 4ac = 900 - 800 = 100 \] Now, we can substitute back into the quadratic formula: \[ x = \frac{-30 \pm \sqrt{100}}{2 \times 8} = \frac{-30 \pm 10}{16} \] This gives us two solutions: 1. \( x = \frac{-30 + 10}{16} = \frac{-20}{16} = -\frac{5}{4} \) 2. \( x = \frac{-30 - 10}{16} = \frac{-40}{16} = -\frac{5}{2} \) Thus, the solutions to the equation are \( x = -\frac{5}{4} \) and \( x = -\frac{5}{2} \). For a little historical context, the quadratic formula has roots tracing back to ancient civilizations, notably the Babylonians around 2000 BC, who solved quadratic equations geometrically long before the formula was formalized. Isn't it fascinating how concepts such as solving for roots were understood so early? Now, in terms of real-world application, quadratic equations pop up everywhere! From calculating projectile motion to optimizing profits in business situations, understanding how to solve them can be immensely helpful in fields like physics, economics, and engineering. Knowing how to apply these equations in practical settings can really boost your problem-solving toolkit!