a
x
2
+
b
x
+
c
=
0
a
x
2
+
b
x
=
−
c
x
2
+
b
a
x
=
−
c
a
Divide out leading coefficient.
x
2
+
b
a
x
+
(
b
2
a
)
2
=
−
c
(
4
a
)
a
(
4
a
)
+
b
2
4
a
2
Complete the square.
(
x
+
b
2
a
)
(
x
+
b
2
a
)
=
b
2
−
4
a
c
4
a
2
Discriminant revealed.
(
x
+
b
2
a
)
2
=
b
2
−
4
a
c
4
a
2
x
+
b
2
a
=
b
2
−
4
a
c
4
a
2
x
=
−
b
2
a
±
{
C
}
b
2
−
4
a
c
4
a
2
There's the vertex formula.
x
=
−
b
±
{
C
}
b
2
−
4
a
c
2
a
\begin{aligned}
ax^2 + bx + c &= 0 \\
ax^2 + bx &= -c \\
x^2 + \frac{b}{a}x &= -\frac{c}{a} & \text{\color{red} \small Divide out leading coefficient.} \\
x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 &= \frac{-c(4a)}{a(4a)} + \frac{b^2}{4a^2} & \text{\color{red} \small Complete the square.} \\
\left(x + \frac{b}{2a}\right)\left(x + \frac{b}{2a}\right) &= \frac{b^2 - 4ac}{4a^2} & \text{\color{red} \small Discriminant revealed.} \\
\left(x + \frac{b}{2a}\right)^2 &= \frac{b^2 - 4ac}{4a^2} \\
x + \frac{b}{2a} &= \sqrt{\frac{b^2 - 4ac}{4a^2}} \\
x &= \frac{-b}{2a} \pm {C} \sqrt{\frac{b^2 - 4ac}{4a^2}} & \text{\color{red} \small There's the vertex formula.} \\
x &= \frac{-b \pm {C}\sqrt{b^2 - 4ac}}{2a}
\end{aligned}