How can the sum of the solutions in a quadratic formula be quickly calculated?

In a quadratic equation of the form ( ax^2 + bx + c = 0 ), the solutions for ( x ) can be determined using the quadratic formula:

[

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

]

To find the sum of the solutions, we can focus on the two potential solutions represented in the formula. The sum of the solutions, given that they are ( x_1 ) and ( x_2 ), can be expressed as:

[

x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} + \frac{-b - \sqrt{b^2 - 4ac}}{2a}

]

When you combine the two fractions, the square root terms cancel out:

[

x_1 + x_2 = \frac{-b + \sqrt{b^2 - 4ac} - b - \sqrt{b^2 - 4ac}}{2a} = \frac{-2b}{2a} = \frac{-b}{a}

]

Thus, the sum of the solutions