solve the following quadratic equations omplete solution. \( 10 x^{2}+5 x=0 \)

To solve the quadratic equation \(10x^2 + 5x = 0\), start by factoring out the common term: \[ 5x(2x + 1) = 0 \] Now, set each factor equal to zero: 1. \(5x = 0\) which gives \(x = 0\), 2. \(2x + 1 = 0\) which leads to \(2x = -1\) or \(x = -\frac{1}{2}\). Thus, the solutions to the equation are \(x = 0\) and \(x = -\frac{1}{2}\). Here’s a tip: always check your solutions by plugging them back into the original equation to ensure they satisfy it. Additionally, a common mistake to avoid is forgetting to factor out any common terms before solving, which can lead to missing solutions!