To fill in the missing reason in the given proof of the quadratic formula and arrive at the given numerical result, let's go through the steps in detail:
1. Given:
[tex]\[
ax^2 + bx + c = 0
\][/tex]
2. Subtract [tex]\( c \)[/tex] from both sides of the equation:
[tex]\[
ax^2 + bx = -c
\][/tex]
3. Divide both sides of the equation by [tex]\( a \)[/tex]:
[tex]\[
x^2 + \frac{b}{a}x = -\frac{c}{a}
\][/tex]
4. Complete the square and add [tex]\(\left(\frac{b}{2a}\right)^2\)[/tex] to both sides:
[tex]\[
x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2
\][/tex]
5. Square [tex]\(\left(\frac{b}{2a}\right)\)[/tex] on the right side of the equation:
[tex]\[
x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}
\][/tex]
6. Find a common denominator on the right side of 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]
7. Add the fractions together on the right side of the equation:
[tex]\[
x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}
\][/tex]
Therefore, the missing reason is:
[tex]\[
\text{Add the fractions together on the right side of the equation}
\][/tex]
