\[ x^{2}-5 x+6 \] 3) What value of \( g \) will make the expression: \( 4 x^{2}+18 x+g \) coul mave it to have equal roots (perfect sqdare)?

To have equal roots, the discriminant of the quadratic equation must be zero. The discriminant \(\Delta\) for the equation \( 4x^{2} + 18x + g \) is calculated as \(\Delta = b^{2} - 4ac\). Here, \(a = 4\), \(b = 18\), and \(c = g\). So, we can set up the equation: \[ \Delta = 18^{2} - 4 \cdot 4 \cdot g = 0. \] Solving for \(g\), we get: \[ 324 - 16g = 0 \implies 16g = 324 \implies g = \frac{324}{16} = 20.25. \] So, \(g = 20.25\) will make \(4x^{2} + 18x + g\) have equal roots, turning it into a perfect square! Now let’s explore how quadratics like this are actually found in many real-life scenarios! For example, the trajectory of a ball thrown in the air can be modeled using quadratic equations. The highest point – or vertex – of that curve can be represented by adjusting the coefficients just right, which is what you do when you're finding those roots. Quadratics pop up in everything from sports to finance—neatly illustrating how math helps us understand the world!