To find the missing steps of the derivation for the quadratic formula, we'll follow these steps:
1. Given Equation:
[tex]\[
a x^2 + b x + c = 0
\][/tex]
2. Isolate [tex]\( c \)[/tex]:
[tex]\[
a x^2 + b x = -c
\][/tex]
3. Convert to simpler quadratic form by dividing through by [tex]\( a \)[/tex]:
[tex]\[
x^2 + \frac{b}{a} x = -\frac{c}{a}
\][/tex]
4. Complete the square on the left side:
[tex]\[
x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2
\][/tex]
5. Simplify and write the left side as a perfect square:
[tex]\[
\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}
\][/tex]
6. Take the square root of both sides:
[tex]\[
x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\][/tex]
7. Solve for [tex]\( x \)[/tex] by isolating it:
[tex]\[
x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\][/tex]
So the missing steps are:
- Step 3:
[tex]\[
x^2 + \frac{b}{a} x = -\frac{c}{a}
\][/tex]
- Step 5:
[tex]\[
\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}
\][/tex]
- Step 8:
[tex]\[
x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}
\][/tex]
So, the steps now look like this:
1. [tex]\( a x^2 + b x + c = 0 \)[/tex]
2. [tex]\( a x^2 + b x = -c \)[/tex]
3. [tex]\( x^2 + \frac{b}{a} x = -\frac{c}{a} \)[/tex]
4. [tex]\( x^2 + \frac{b}{a} x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2} \)[/tex]
5. [tex]\(\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \)[/tex]
6. [tex]\( x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \)[/tex]
7. [tex]\( x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} \)[/tex]
These missing steps complete the derivation of the quadratic formula.
