A quadratic equation is given in the form $$ax^2 + bx + c = 0$$. The equation is $$-4x^2 + 19x - 12 = 0$$. Identify the values of a, b, and c. Then, calculate the discriminant of the equation using the formula $$D = b^2 - 4ac$$. Finally, calculate the roots of the quadratic equation using the formula $$x = \frac{-b \pm \sqrt{D}}{2a}$$.

Solution:

• Given equation: $$-4x^2 + 19x - 12 = 0$$
• Identifying coefficients: From the equation, we have:
  • $$a = -4$$
  • $$b = 19$$
  • $$c = -12$$
• Calculating the discriminant (D): Using the formula $$D = b^2 - 4ac$$:
  • $$D = (19)^2 - 4(-4)(-12)$$
  • $$D = 361 - 192$$
  • $$D = 169$$
• Calculating the roots (x): Using the formula $$x = \frac{-b \pm \sqrt{D}}{2a}$$:
  • $$x_1 = \frac{-19 + \sqrt{169}}{2(-4)} = \frac{-19 + 13}{-8} = \frac{-6}{-8} = \frac{3}{4}$$
  • $$x_2 = \frac{-19 - \sqrt{169}}{2(-4)} = \frac{-19 - 13}{-8} = \frac{-32}{-8} = 4$$

Answer: The values are $$a = -4$$, $$b = 19$$, $$c = -12$$. The discriminant is $$D = 169$$. The roots of the equation are $$x_1 = \frac{3}{4}$$ and $$x_2 = 4$$.