Show that the roots of the following equation are real: \[ 2 x^{2}+3 x-3=6 x^{2}-8 x+4 \] Answer: \( \Delta=\square \); therefore the equation has two real roots.

First, let's simplify the equation. Moving all terms to one side, we get: \[ 2x^2 + 3x - 3 - 6x^2 + 8x - 4 = 0 \] Which simplifies to: \[ -4x^2 + 11x - 7 = 0 \] Now, to apply the quadratic formula and check for real roots, we need the discriminant, \( \Delta \): \[ \Delta = b^2 - 4ac = (11)^2 - 4(-4)(-7) = 121 - 112 = 9 \] Since \( \Delta = 9 \) is greater than zero, we conclude that the equation has two distinct real roots. Real-world applications of quadratic equations can be found in various fields like physics and engineering, where projectile motion and structural analysis often involve such equations. For instance, the path of a thrown ball can be modeled using a quadratic equation that accounts for height versus time—talk about applying math in real life! When solving quadratic equations, a common mistake is forgetting to simplify the equation properly before calculating the discriminant. Ensure to always combine like terms or move all terms to one side first! It can also be easy to overlook that a negative discriminant indicates complex roots; so double-checking the computations is key to avoiding confusion.