Deriving the Quadratic Formula
The standard form of the quadratic equation is \(ax^2+bx+c=0\). Use your understanding of completing the square to put the steps below in the correct order to derive the quadratic formula.
| \( \left( x + \dfrac{b}{2a} \right) = \pm \sqrt{ \dfrac{b^2-4ac}{4a^2}}\) |
| \(x^2 + \dfrac{b}{a} x + \bigg( \dfrac{b}{2a} \bigg)^2 = – \dfrac{c}{a}+\bigg( \dfrac{b}{2a}\bigg)^2 \) |
| \( \sqrt{\bigg( x + \dfrac{b}{2a}\bigg)^2} = \pm \sqrt{ \dfrac{b^2-4ac}{4a^2}}\) |
| \(x = \dfrac {-b \pm \sqrt{b^2-4ac} } {2a}\) |
| \( \bigg( x + \dfrac{b}{2a} \bigg)^2 = -\dfrac{c}{a} \cdot \dfrac{4a}{4a} + \dfrac{b^2}{4a^2}\) |
| \( x + \dfrac{b}{2a} = \pm \dfrac{\sqrt{ b^2-4ac}}{2a} \) |
| \(x^2 + \dfrac{b}{a} x + \dfrac{c}{a} = 0\), \(a \neq 0 \) |
| \( x = -\dfrac{b}{2a} \pm \dfrac{\sqrt{b^2-4ac}}{2a}\) |
| \( \bigg( x + \dfrac{b}{2a}\bigg)^2 = -\dfrac{c}{a}+\bigg( \dfrac{b}{2a}\bigg)^2 \) |
| \( x + \dfrac{b}{2a} = \pm \dfrac{\sqrt{b^2-4ac}}{ \sqrt{4a^2}}\) |
| \(\bigg( x + \dfrac{b}{2a}\bigg)^2 = -\dfrac{c}{a} + \dfrac{b^2}{4a^2}\) |
| \(x^2 + \dfrac{b}{a} x = \ – \dfrac{c}{a} \) |
| \(\bigg( x + \dfrac{b}{2a}\bigg)^2 = \dfrac{b^2-4ac}{4a^2}\) |
