Where did quadratic formula came from

Mathematics

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Answer

The quadratic formula is derived by applying the method of completing the square to the general quadratic equation ax2+bx+c=0ax^2 + bx + c = 0.

Step-by-step solution

9 steps

  1. Start with the general quadratic equation.

    ax2+bx+c=0ax^2 + bx + c = 0

  2. Divide the entire equation by aa (assuming a≠0a \neq 0).

    x2+bax+ca=0x^2 + \frac{b}{a}x + \frac{c}{a} = 0

  3. Move the constant term to the right side of the equation.

    x2+bax=−cax^2 + \frac{b}{a}x = -\frac{c}{a}

  4. Complete the square on the left side.

    To do this, add (b2a)2(\frac{b}{2a})^2 to both sides of the equation.

    x2+bax+(b2a)2=−ca+(b2a)2x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2

  5. Factor the left side as a perfect square and simplify the right side.

    (x+b2a)2=−ca+b24a2\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2}

  6. Combine the terms on the right side by finding a common denominator, which is 4a24a^2.

    (x+b2a)2=−4ac4a2+b24a2\left(x + \frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} (x+b2a)2=b2−4ac4a2\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}

  7. Take the square root of both sides.

    Remember to include both positive and negative roots.

    x+b2a=±b2−4ac4a2x + \frac{b}{2a} = \pm\sqrt{\frac{b^2 - 4ac}{4a^2}} x+b2a=±b2−4ac4a2x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{\sqrt{4a^2}} x+b2a=±b2−4ac2ax + \frac{b}{2a} = \pm\frac{\sqrt{b^2 - 4ac}}{2a}

  8. Isolate xx by subtracting b2a\frac{b}{2a} from both sides.

    x=−b2a±b2−4ac2ax = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a}

  9. Combine the terms on the right side since they have a common denominator.

    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

    This is the quadratic formula.

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