Certainly! Let's delve into the detailed solution step-by-step.
1. Combine the fractions on the right side:
Given the equation:
[tex]\[
x^2 + \frac{b}{a} x + \left( \frac{b}{2a} \right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2}
\][/tex]
We want a common denominator to combine the fractions. The common denominator is [tex]\( 4a^2 \)[/tex]:
[tex]\[
-\frac{4ac}{4a^2} + \frac{b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2}
\][/tex]
2. Add the fractions together on the right side:
After combining the fractions, the equation becomes:
[tex]\[
x^2 + \frac{b}{a} x + \left( \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}
\][/tex]
3. Rewrite the left side as a perfect square trinomial:
Notice that the left side of the equation is a perfect square trinomial. It can be expressed as a squared binomial:
[tex]\[
\left( x + \frac{b}{2a} \right)^2 = \frac{b^2 - 4ac}{4a^2}
\][/tex]
4. Take the square root of both sides:
To solve for [tex]\( x \)[/tex], we take the square root of both sides. Remember that when you take the square root of both sides, you should consider both the positive and negative roots:
[tex]\[
x + \frac{b}{2a} = \pm \sqrt{ \frac{b^2 - 4ac}{4a^2} }
\][/tex]
5. Simplify the square root:
The square root on the right side can be simplified:
[tex]\[
\sqrt{ \frac{b^2 - 4ac}{4a^2} } = \frac{ \sqrt{b^2 - 4ac} }{2a}
\][/tex]
Thus, the equation becomes:
[tex]\[
x + \frac{b}{2a} = \pm \frac{ \sqrt{b^2 - 4ac} }{2a}
\][/tex]
6. Isolate [tex]\( x \)[/tex] on one side:
Finally, to solve for [tex]\( x \)[/tex], we subtract [tex]\( \frac{b}{2a} \)[/tex] from both sides:
[tex]\[
x = - \frac{b}{2a} \pm \frac{ \sqrt{b^2 - 4ac} }{2a}
\][/tex]
Combine the fractions to get the final form of the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
This is the standard quadratic formula used to solve quadratic equations of the form [tex]\( ax^2 + bx + c = 0 \)[/tex].
Therefore, our final answer is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
