Completing the square is a powerful technique for solving quadratic equations. It transforms a quadratic expression into a perfect square trinomial, making it easier to find solutions. This method is especially useful when dealing with equations that can't be easily factored.

By mastering completing the square, you'll gain a deeper understanding of quadratic relationships. It's not just about finding solutions – it also reveals the vertex form of quadratic functions, which is crucial for graphing and analyzing parabolas.

Completing the Square

Binomial to perfect square trinomial

• Binomial expression in the form $ax^2 + bx$
  • $a$ represents coefficient of $x^2$ term
  • $b$ represents coefficient of $x$ term
• Transform binomial into perfect square trinomial by adding square of half the coefficient of $x$ to both sides of equation
  • Calculate $(\frac{b}{2})^2$ and add to both sides
  • Resulting trinomial on left side will be perfect square in the form $(ax + \frac{b}{2})^2$
  • Example: $x^2 + 6x$ becomes $(x + 3)^2$ by adding $(\frac{6}{2})^2 = 3^2 = 9$ to both sides

Completing the square with coefficient 1

• Quadratic equation in standard form $x^2 + bx + c = 0$, where leading coefficient $a = 1$
• Isolate variable terms on one side by subtracting $c$ from both sides
  • $x^2 + bx = -c$
• Transform binomial $x^2 + bx$ into perfect square trinomial
  • Add $(\frac{b}{2})^2$ to both sides of equation
  • Left side becomes $(x + \frac{b}{2})^2 = -c + (\frac{b}{2})^2$
• Take square root of both sides
  • $x + \frac{b}{2} = \pm \sqrt{-c + (\frac{b}{2})^2}$
• Solve for $x$ by subtracting $\frac{b}{2}$ from both sides
  • $x = -\frac{b}{2} \pm \sqrt{-c + (\frac{b}{2})^2}$
  • Example: $x^2 + 6x + 5 = 0$ becomes $x = -3 \pm \sqrt{-5 + (\frac{6}{2})^2} = -3 \pm \sqrt{4} = -3 \pm 2$, so $x = -5$ or $x = -1$

Completing the square for any coefficient

• Quadratic equation in standard form $ax^2 + bx + c = 0$, where leading coefficient $a \neq 1$
• Divide both sides by $a$ to make leading coefficient 1
  • $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$
• Isolate variable terms on one side by subtracting $\frac{c}{a}$ from both sides
  • $x^2 + \frac{b}{a}x = -\frac{c}{a}$
• Transform binomial $x^2 + \frac{b}{a}x$ into perfect square trinomial
  • Add $(\frac{b}{2a})^2$ to both sides of equation
  • Left side becomes $(x + \frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2$
• Take square root of both sides
  • $x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + (\frac{b}{2a})^2}$
• Solve for $x$ by subtracting $\frac{b}{2a}$ from both sides
  • $x = -\frac{b}{2a} \pm \sqrt{-\frac{c}{a} + (\frac{b}{2a})^2}$
  • Example: $2x^2 + 12x + 7 = 0$ becomes $x = -\frac{12}{2(2)} \pm \sqrt{-\frac{7}{2} + (\frac{12}{2(2)})^2} = -3 \pm \sqrt{\frac{1}{2}} = -3 \pm \frac{\sqrt{2}}{2}$

Other Methods for Solving Quadratic Equations

• Factoring: Useful when the quadratic expression can be easily factored
• Quadratic formula: An alternative method derived from completing the square
• Discriminant: Helps determine the nature of roots (real or complex) without solving the equation
  • For $ax^2 + bx + c = 0$, the discriminant is $b^2 - 4ac$
  • If discriminant > 0, there are two distinct real roots
  • If discriminant = 0, there is one real root (repeated)
  • If discriminant < 0, there are two complex roots