Quadratic Formula

Quadratic equations (\( ax^2 + bx + c = 0 \)) model parabolas and are key in algebra, physics, and economics. At MathMultiverse, we break down the quadratic formula with clear examples, visualizations, and applications.

Formula

Roots of \( ax^2 + bx + c = 0 \):

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

Discriminant (\( b^2 - 4ac \)) determines root types: positive (two real), zero (one real), negative (complex).

Examples

Basic Quadratic

\( x^2 - 5x + 6 = 0 \):

\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} = \frac{5 \pm 1}{2} \] \[ x = 3, 2 \]

Decimals

\( 0.2x^2 + 1.5x - 3 = 0 \):

\[ x = \frac{-1.5 \pm \sqrt{1.5^2 - 4(0.2)(-3)}}{0.4} \approx 1.64, -9.14 \]

Complex Roots

\( x^2 + 2x + 5 = 0 \):

\[ x = \frac{-2 \pm \sqrt{4 - 20}}{2} = -1 \pm 2i \]

Word Problem

Height \( h = -5t^2 + 20t + 2 \), find ground impact:

\[ t = \frac{20 \pm \sqrt{440}}{10} \approx 4.10 \text{ seconds} \]

Discriminant Analysis

\( 2x^2 + 3x - 5 = 0 \):

\[ x = \frac{-3 \pm \sqrt{49}}{4} = 1, -2.5 \]

Optimization

Maximize \( A = -x^2 + 10x \):

\[ x = -\frac{10}{2(-1)} = 5, \quad A = 25 \]

Derivation

From \( ax^2 + bx + c = 0 \), complete the square:

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

Visualizations

\( y = x^2 - 5x + 6 \)

\( y = x^2 + 2x + 5 \)

Applications

  • Projectile Motion: \( h = -16t^2 + 40t \), ground at \( t = 2.5 \) seconds.
  • Profit: \( P = -2x^2 + 100x - 500 \), max $750 at \( x = 25 \).
  • Area: \( A = -x^2 + 50x \), max 625 m² at \( x = 25 \).
  • Engineering: \( d = 0.01x^2 - 0.5x + 5 \), zero at \( x \approx 13.85, 36.15 \).
  • Economics: \( C = 0.03x^2 - 2x + 100 \), no real break-even.
  • Agriculture: \( Y = -0.01x^2 + 4x \), max 400 tons at 200 acres.