Solve the quadratic equation by factoring. \[ 3-22 m=-7 m^{2} \]

First, we rearrange the equation to bring all terms to one side: \[ 7m^2 - 22m + 3 = 0. \] Next, we need to factor this quadratic. We are looking for two numbers that multiply to \( 7 \times 3 = 21 \) and add to \( -22 \). The numbers are -21 and -1. Now we can rewrite the middle term using these numbers: \[ 7m^2 - 21m - 1m + 3 = 0. \] Next, we group the terms: \[ (7m^2 - 21m) + (-1m + 3) = 0. \] Factoring out common factors gives us: \[ 7m(m - 3) - 1(m - 3) = 0. \] Now, we factor out \( (m - 3) \): \[ (m - 3)(7m - 1) = 0. \] Setting each factor to zero gives us: 1. \( m - 3 = 0 \) (thus, \( m = 3 \)) 2. \( 7m - 1 = 0 \) (thus, \( m = \frac{1}{7} \)) So, the solutions are \( m = 3 \) and \( m = \frac{1}{7} \). Now let's add some engaging context! Ever wondered how long ago people were solving quadratic equations just like this? Quadratics have been studied since ancient civilizations, with the Babylonians dabbling in algebra around 2000 BC. They had methods resembling today's techniques, albeit without modern symbols! Now, in a real-world application, quadratic equations pop up everywhere—like modeling the trajectory of a basketball shot! Have you ever shot a hoop? The path of that ball can be represented using a quadratic equation. So, the next time you're shooting some hoops, remember that math is there helping you find the perfect angle!