Solving Quadratic Equations Using the Quadratic Formula page 11

Overview of the Quadratic Formula

  • Definition: The quadratic formula is a universal algebraic formula used to solve any quadratic equation in standard form.

  • Method Classification: It represents the fourth method of solving quadratic equations.

  • Standard Form Requirement: The equation must be written in the standard form ax2+bx+c=0a x^2 + b x + c = 0, where a≠0a \neq 0.

  • Practical Guidelines and Common Pitfalls:

    • While applicable to any quadratic equation, errors frequently occur when substituting values into the formula.

    • Precise attention must be paid during the substitution process.

    • Parentheses should always be used when inserting a negative number into the formula to prevent sign errors.

Derivation of the Quadratic Formula by Completing the Square

  • Mathematical Foundation: The quadratic formula is derived directly by completing the square on the general quadratic equation ax2+bx+c=0a x^2 + b x + c = 0 (where a≠0a \neq 0).

  • Leading Coefficient Adjustment:

    • The derivation assumes that the leading coefficient aa is positive (a>0a > 0).

    • If the leading coefficient aa is negative, multiply the entire equation by −1-1 to obtain a positive leading coefficient aa.

  • Step 1: Isolate the Constant Term

    • Move the constant term cc to the right side of the equal sign: ax2+bx=−ca x^2 + b x = -c

  • Step 2: Make the Leading Coefficient Equal to 1

    • Divide through the entire equation by aa: x2+bax=−cax^2 + \frac{b}{a} x = -\frac{c}{a}

  • Step 3: Complete the Square

    • Calculate 12\frac{1}{2} of the middle term's coefficient (ba\frac{b}{a}): 12⋅ba=b2a\frac{1}{2} \cdot \frac{b}{a} = \frac{b}{2a}

    • Square this value to determine the term needed to complete the square: (12⋅ba)2=b24a2\left(\frac{1}{2} \cdot \frac{b}{a}\right)^2 = \frac{b^2}{4a^2}

    • Add b24a2\frac{b^2}{4a^2} to both sides of the equal sign: x2+bax+b24a2=b24a2−cax^2 + \frac{b}{a} x + \frac{b^2}{4a^2} = \frac{b^2}{4a^2} - \frac{c}{a}

  • Step 4: Rewrite as a Perfect Square and Combine Terms

    • Express the left side as a perfect square: (x+b2a)2\left(x + \frac{b}{2a}\right)^2

    • Find the common denominator (4a24a^2) for the right side and write it as a single fraction: (x+b2a)2=b2−4ac4a2\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}

  • Step 5: Apply the Square Root Property

    • Apply the square root property to both sides: x+b2a=±b2−4ac4a2x + \frac{b}{2a} = \pm \sqrt{\frac{b^2 - 4ac}{4a^2}}

    • Simplify the denominator under the square root: x+b2a=±b2−4ac2ax + \frac{b}{2a} = \frac{\pm \sqrt{b^2 - 4ac}}{2a}

  • Step 6: Isolate the Variable x

    • Add −b2a-\frac{b}{2a} to both sides of the equation and combine the terms on the right side over the common denominator 2a2a: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}