Completing the Square: Quick Reference

  • Standard form and case distinction
    • Quadratic in standard form: ax^2 + bx + c = 0
    • If a = 1, proceed directly to completing the square. If a \neq 1, divide the entire equation by a to make the leading coefficient 1.
    • After making leading coefficient 1: x^2 + \frac{b}{a}x + \frac{c}{a} = 0
  • Completing the square: core steps
    • Move the constant to the other side: x^2 + \frac{b}{a}x = -\frac{c}{a}
    • Take half of the linear coefficient: \frac{b}{2a}
    • Square it: \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}
    • Add the square to both sides to form a trinomial square on the left:
      \left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} = \frac{b^2 - 4ac}{4a^2}
    • Solve by square root:
      x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a}
    • Isolate x:
      x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • Example Set 1: a = 1 (monic equations)
    • Problem 1: x^2 - 4x - 14 = 0
    • Move constant: x^2 - 4x = 14
    • Half of -4: -2; square: 4; add both sides: x^2 - 4x + 4 = 14 + 4
    • Binomial square: \left(x - 2\right)^2 = 18
    • Square root: x - 2 = \pm \sqrt{18} = \pm 3\sqrt{2}
    • Solutions: x = 2 \pm 3\sqrt{2}
    • Problem 2: x^2 - 4x + 2 = 0
    • Move: x^2 - 4x = -2
    • Add: \left(x - 2\right)^2 = -2 + 4 = 2
    • Solve: x = 2 \pm \sqrt{2}
    • Problem 3: x^2 - 7x + 12 = 0
    • Move: x^2 - 7x = -12
    • Half of -7: -\frac{7}{2}; square: \frac{49}{4}; add: \left(x - \frac{7}{2}\right)^2 = -12 + \frac{49}{4} = \frac{1}{4}
    • Square root: x - \frac{7}{2} = \pm \frac{1}{2}
    • Solutions: x = \frac{7}{2} \pm \frac{1}{2} \Rightarrow x = 4 \text{ or } x = 3
  • Example Set 2: a ≠ 1 (divide by a first)
    • Problem 4: 2x^2 - x - 28 = 0
    • Divide by 2: x^2 - \frac{1}{2}x - 14 = 0
    • Move: x^2 - \frac{1}{2}x = 14
    • Half of the middle term: -\frac{1}{4}; square: \frac{1}{16}; add both sides: \left(x - \frac{1}{4}\right)^2 = 14 + \frac{1}{16} = \frac{225}{16}
    • Square root: x - \frac{1}{4} = \pm \frac{15}{4}
    • Solutions: x = \frac{1}{4} \pm \frac{15}{4} \Rightarrow x = 4 \text{ or } x = -\frac{7}{2}
    • Problem 5: 4x^2 - 3x - 7 = 0
    • Divide by 4: x^2 - \frac{3}{4}x - \frac{7}{4} = 0
    • Move: x^2 - \frac{3}{4}x = \frac{7}{4}
    • Half of -\frac{3}{4}: -\frac{3}{8}; square: \frac{9}{64}; add: \left(x - \frac{3}{8}\right)^2 = \frac{7}{4} + \frac{9}{64} = \frac{121}{64}
    • Square root: x - \frac{3}{8} = \pm \frac{11}{8}
    • Solutions: x = \frac{3}{8} \pm \frac{11}{8} \Rightarrow x = \frac{7}{4} \text{ or } x = -1
  • Quick reminder on fractions when completing the square
    • Additions on both sides use a common denominator when fractions are involved. For example, convert constants to the same denominator before summing.
  • Takeaways
    • If a = 1, proceed directly with completing the square. If a ≠ 1, first divide by a to normalize to a = 1, then complete the square.
    • Completing the square always yields a square of a binomial on the left, leading to a straightforward square-root step and two solutions (when real).