11. \( 4 x^{2}=x+3 \)

To solve the equation \( 4x^{2} = x + 3 \), we first rearrange it into standard form by moving all terms to one side: \[ 4x^{2} - x - 3 = 0. \] Now, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a = 4 \), \( b = -1 \), and \( c = -3 \). Plugging in these values, we get: \[ x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4(4)(-3)}}{2(4)} = \frac{1 \pm \sqrt{1 + 48}}{8} = \frac{1 \pm \sqrt{49}}{8} = \frac{1 \pm 7}{8}. \] Thus, \( x = 1 \) or \( x = -\frac{3}{4} \). Now, let's explore the fascinating journey of quadratic equations through time! Did you know that quadratic equations have roots in ancient civilizations? The Babylonians, as early as 2000 BC, were solving quadratic problems without the formal structure we know today. They utilized geometric methods to find the roots, paving the way for algebraic breakthroughs! Who knew mathematics could have such a storied past? And for those who love applying math in the real world, quadratic equations pop up all over the place—like in projectile motion! Think about throwing a ball: the height of the ball can be modeled with a quadratic equation. So next time you toss that ball, you’re not just playing; you’re playing with math in action! How cool is that?