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Quadratic Formula Derivation

Video Title:-Quadratic Formula Derivation The quadratic formula is derived from the process of completing the square for a general quadratic equation of the form ( ax^2 + bx + c = 0 ). Here's a step-by-step outline of the derivation:Start with the General Quadratic Equation: [ ax^2 + bx + c = 0 ]Divide through by ( a ) (assuming ( a
eq 0 )) to simplify: [ x^2 + \frac{b}{a}x + \frac{c}{a} = 0 ]Rearrange the equation to isolate the constant term on the right side: [ x^2 + \frac{b}{a}x = -\frac{c}{a} ]Complete the square on the left side:To complete the square, add and subtract ( \left(\frac{b}{2a}\right)^2 ) inside the equation: [ x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 ]This simplifies to: [ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} ]Take the square root of both sides: [ x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}} ] [ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} ]Solve for ( x ) by isolating it: [ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} ] [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]The result, ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), is the quadratic formula. It provides the solutions to any quadratic equation of the form ( ax^2 + bx + c = 0 ).