08Bquadraticformulaproof

= Proof of the Quadratic Formula =

The quadratic formula (shown on the right) can be obtained from the standard quadratic equation:

... ... ... ** ax 2 + bx + c = 0 **

The processes uses completing the square.

math \\ . \qquad ax^2 + bx + c = 0 \qquad \qquad \qquad \;\; \big\{ \text{divide through by } a \big\} \\. \\ . \qquad x^2 + \dfrac{b}{a}x + \dfrac{c}{a} = 0 \qquad \qquad \qquad \big\{ \text{complete the square} \big\} \\. \\ . \qquad x^2 + \dfrac{b}{a}x + \left( \dfrac{b}{2a} \right)^2 - \left( \dfrac{b}{2a} \right)^2 + \dfrac{c}{a} = 0 math . math \\ . \qquad \left( x + \dfrac{b}{2a} \right)^2 - \dfrac{b^2}{4a^2} + \dfrac{c}{a} = 0 \qquad \qquad \big\{ \text{multiply last term by } \dfrac{4a}{4a} \big\} \\. \\ . \qquad \left( x + \dfrac{b}{2a} \right)^2 - \left( \dfrac{b^2}{4a^2} - \dfrac{4ac}{4a^2} \right) = 0 \\. \\ . \qquad \left( x + \dfrac{b}{2a} \right)^2 - \dfrac{b^2-4ac}{4a^2} = 0 \qquad \qquad \big\{ \text{add 2nd term to both sides} \big\} math . math \\ . \qquad \left( x + \dfrac{b}{2a} \right)^2 = \dfrac{b^2-4ac}{4a^2} \qquad \qquad \quad \; \big\{ \text{square root both sides} \big\} \\. \\ . \qquad \; \; x + \dfrac{b}{2a} = \pm \sqrt{ \dfrac{b^2-4ac}{4a^2} } \\. \\ . \qquad \; \; x + \dfrac{b}{2a} = \pm \dfrac{ \sqrt{b^2-4ac} }{2a} \qquad \qquad \; \big\{ \text{subtract } \dfrac{b}{2a} \text{ from both sides} \big\} \\. \\ . \qquad \; \; x = -\dfrac{b}{2a} \pm \dfrac{ \sqrt{b^2-4ac} }{2a} \\. \\ . \qquad \; \; x = \dfrac{ -b \pm \sqrt{b^2-4ac} }{2a} math

And that is the quadratic formula!!