How do you derive the quadratic formula from a standard quadratic equation?

Answers:

• To derive the quadratic formula from the standard quadratic equation, we start with the general form of a quadratic equation, which is:

  [ ax^2 + bx + c = 0 ]

  where ( a ), ( b ), and ( c ) are constants, and ( a \neq 0 ).

  The goal is to solve for ( x ) in this equation. We'll use the method of completing the square to derive the quadratic formula.

  Step 1: Divide by ( a )

  First, divide the entire equation by ( a ) (assuming ( a \neq 0 )):

  [ x^2 + \frac{b}{a} x + \frac{c}{a} = 0 ]

  Step 2: Rearrange the equation

  Next, rearrange the equation to isolate the constant term on one side:

  [ x^2 + \frac{b}{a} x = -\frac{c}{a} ]

  Step 3: Complete the square

  Now, we complete the square on the left side. To do this, take half of the coefficient of ( x ), square it, and add it to both sides. The coefficient of ( x ) is ( \frac{b}{a} ):

  1. Half of ( \frac{b}{a} ) is ( \frac{b}{2a} ).
  2. Squaring this gives ( \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} ).

  Add this square to both sides of the equation:

  [ x^2 + \frac{b}{a} x + \frac{b^2}{4a^2} = -\frac{c}{a} + \frac{b^2}{4a^2} ]

  Now, the left side can be factored as a square:

  [ \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} ]

  Step 4: Simplify the right side

  To combine the terms on the right side, we need a common denominator. The common denominator is ( 4a^2 ):

  [ -\frac{c}{a} = -\frac{4ac}{4a^2} ]

  Thus, we can rewrite the right side:

  [ \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} ]

  Step 5: Take the square root of both sides

  Taking the square root of both sides gives:

  [ x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} ]

  Step 6: Solve for ( x )

  Now, isolate ( x ):

  [ x = -\frac{b}{2a} \pm \frac{\sqrt{b^2 - 4ac}}{2a} ]

  Combining the terms on the right side:

  [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

  Final Result

  The quadratic formula is:

  [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

  This formula provides the solutions for any quadratic equation in standard form ( ax^2 + bx + c = 0 ).