The problem asks for the general formula to obtain the roots of a quadratic equation of the form $ax^2 + bx + c = 0$.

1. Problem Description

The problem asks for the general formula to obtain the roots of a quadratic equation of the form $ax^2 + bx + c = 0$.

2. Solution Steps

The quadratic formula is derived by completing the square on the quadratic equation $ax^2 + bx + c = 0$.

First, divide the equation by $a$ (assuming $a \ne 0$):

$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

Next, move the constant term to the right side:

$x^2 + \frac{b}{a}x = -\frac{c}{a}$

Now, complete the square on the left side. To do this, take half of the coefficient of the $x$ term, square it, and add it to both sides. Half of $\frac{b}{a}$ is $\frac{b}{2a}$, and its square is $(\frac{b}{2a})^2 = \frac{b^2}{4a^2}$.

$x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2}$

Rewrite the left side as a perfect square:

$(x + \frac{b}{2a})^2 = \frac{b^2}{4a^2} - \frac{c}{a}$

Find a common denominator on the right side:

$(x + \frac{b}{2a})^2 = \frac{b^2 - 4ac}{4a^2}$

Take the square root of both sides:

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

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

Isolate $x$:

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

Combine the fractions:

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

Therefore, the quadratic formula is:

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

3. Final Answer

The final answer is d. $x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$