Quadratic Formula Explained

Quadratic equations are polynomials of degree 2, written as \( ax^2 + bx + c = 0 \). The quadratic formula provides a direct way to find the roots (solutions) of these equations, even when factoring isn’t obvious.

The Quadratic Formula

The formula is:

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

Where:

  • \(a\): Coefficient of \(x^2\)
  • \(b\): Coefficient of \(x\)
  • \(c\): Constant term

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

Identify coefficients: \(a = 1\), \(b = -5\), \(c = 6\).

Substitute into the formula:

\[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)} \]
\[ x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2} = \frac{5 \pm 1}{2} \]

Solutions: \( x = \frac{6}{2} = 3 \) or \( x = \frac{4}{2} = 2 \).

Derivation (Completing the Square)

Start with \( ax^2 + bx + c = 0 \):

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

Move \( \frac{c}{a} \): \( x^2 + \frac{b}{a}x = -\frac{c}{a} \).

Add \( \left(\frac{b}{2a}\right)^2 \) to both sides:

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

Take square root and solve: \( x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \), leading to the formula.

Graphical View

The roots are where \( y = x^2 - 5x + 6 \) crosses the x-axis:

Applications

Quadratics model projectile motion, profit maximization, and more. For example, the height of a ball thrown upward follows a parabolic path solvable with this formula.