Answer:
[tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
Step-by-step explanation:
Hello!
Let's start with the given equation and work our way towards x.
Solve for x
- [tex]ax^2 + bx + c = 0[/tex]
Divide both sides by a
- [tex]a(x^2 + \frac bax + \frac ca) = 0[/tex]
- [tex]x^2 + \frac bax + \frac ca = 0[/tex]
Move c/a to the other side
- [tex]x^2 + \frac bax = -\frac ca[/tex]
At the step, we have to use the Completing the Square method.
- [tex]x^2 + \frac bax + (\frac{b}{2a})^2 = -\frac ca + (\frac b{2a})^2[/tex]
- [tex]x^2 + \frac bax + \frac{b^2}{4a^2} = -\frac ca + \frac {b^2}{4a^2}[/tex]
- [tex](x + \frac b{2a})^2= -\frac ca + \frac{b^2}{4a^2}[/tex]
Multiply -c/a by 4a and add the two fractions
- [tex](x + \frac b{2a})^2= \frac{b^2 - 4ac}{4a^2}[/tex]
Square Root both sides
- [tex]\sqrt{(x + \frac b{2a})^2}= \sqrt{\frac{b^2 - 4ac}{4a^2}}[/tex]
- [tex]x + \frac b{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}}[/tex]
Subtract b/2a
- [tex]x = \pm \frac{ \sqrt{b^2 - 4ac}}{2a} - \frac b{2a}[/tex]
- [tex]x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex]
And that's the derivation of the Quadratic Formula.
