Mathematics

A quadratic equation is a second-order polynomial equation in a single variable $x$ of the form:

$a x^2 + b x + c = 0$

Where $a \neq 0$ (if $a = 0$, the equation degrades into a linear equation $b x + c = 0$). The constants $a$, $b$, and $c$ are known as coefficients.

Geometrical Interpretation: Parabola Intercepts

Graphically, the quadratic function $y = a x^2 + b x + c$ represents a parabola. The solutions (roots) of $a x^2 + b x + c = 0$ correspond to the $x$-intercepts where the parabola intersects the horizontal $x$-axis.

1. Discriminant $\Delta > 0$: Parabola crosses the $x$-axis twice $\to$ Two distinct real roots.
2. Discriminant $\Delta = 0$: Parabola vertex touches the $x$-axis $\to$ One repeated real root (double root).
3. Discriminant $\Delta < 0$: Parabola does not cross the $x$-axis $\to$ Two complex conjugate roots ($a \pm b i$).

For calculating percentage variations in algebraic parameters, see Percentage Calculator.

1. Canonical Quadratic Formula

The general solution for the roots of $a x^2 + b x + c = 0$ ($a \neq 0$) is given by:

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

2. The Discriminant ($\Delta$ or $D$)

The term under the square root is the discriminant:

$\Delta = b^2 - 4ac$

Root Classification Matrix

• Case 1: $\Delta > 0$ (Two distinct real roots) $x_1 = \frac{-b + \sqrt{\Delta}}{2a}, \quad x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

• Case 2: $\Delta = 0$ (One real double root) $x_1 = x_2 = -\frac{b}{2a}$

• Case 3: $\Delta < 0$ (Two complex conjugate roots) $x_{1,2} = -\frac{b}{2a} \pm i \frac{\sqrt{-\Delta}}{2a}$

3. Vieta’s Formulas (Root-Coefficient Relations)

For roots $x_1$ and $x_2$: $x_1 + x_2 = -\frac{b}{a}$ $x_1 \cdot x_2 = \frac{c}{a}$

Problem Statement

Solve the quadratic equation:

$2 x^2 - 4 x + 5 = 0$

Identify coefficients $a$, $b$, and $c$, evaluate the discriminant $\Delta$, and find the exact complex roots.

Step-by-Step Numerical Solution

Step 1: Identify Coefficients

• $a = 2$
• $b = -4$
• $c = 5$

Step 2: Compute Discriminant ($\Delta = b^2 - 4ac$)

$\Delta = (-4)^2 - 4(2)(5) = 16 - 40 = -24$

Since $\Delta = -24 < 0$, the equation has two complex conjugate roots.

Step 3: Apply Quadratic Formula for Complex Roots

$x = \frac{-(-4) \pm \sqrt{-24}}{2(2)} = \frac{4 \pm \sqrt{24} \cdot i}{4}$

Simplify $\sqrt{24} = \sqrt{4 \times 6} = 2 \sqrt{6}$: $x = \frac{4 \pm 2 \sqrt{6} i}{4} = 1 \pm \frac{\sqrt{6}}{2} i$

Step 4: Decimal Value Evaluation ($\sqrt{6} \approx 2.4495$)

$x_1 = 1 + 1.2247 i$ $x_2 = 1 - 1.2247 i$

Step 5: Verify using Vieta’s Formula

• Sum: $x_1 + x_2 = 1 + 1 = 2 = -\frac{-4}{2} \checkmark$
• Product: $x_1 \cdot x_2 = 1^2 - (1.2247 i)^2 = 1 + 1.5 = 2.5 = \frac{5}{2} \checkmark$

Result Summary: The roots are $x = 1 \pm 1.225 i$.

Practical Applications

1. Kinematic Projectile Motion Trajectories

   • The height $h(t)$ of a ball or missile launched under gravity follows a quadratic equation: $h(t) = -\frac{1}{2} g t^2 + v_0 t + h_0$. Solving $h(t) = 0$ calculates exact touchdown time.

2. Parabolic Reflector Antenna & Lens Optics

   • Satellite dishes and headlight reflectors utilize parabolic cross-sections ($y = a x^2$). The focal point coordinates rely on the quadratic coefficient $a$ ($f = \frac{1}{4a}$).

3. Profit & Revenue Optimization in Economics

   • Revenue models with price elasticity ($R(p) = -a p^2 + b p$) yield quadratic curves where the vertex gives maximum profit.

Frequently Asked Questions

What is “completing the square”?

Completing the square is an algebraic technique used to rewrite $a x^2 + b x + c = 0$ in the form $a(x - h)^2 + k = 0$. It is the method originally used to derive the general quadratic formula.

What is catastrophic cancellation in floating-point quadratic solvers?

When $b^2 \gg 4ac$, computing $-b + \sqrt{b^2 - 4ac}$ involves subtracting two nearly equal numbers, leading to loss of numerical precision. Professional solvers use Citardauq’s formula: $x_1 = \frac{-2c}{b + \text{sgn}(b) \sqrt{b^2 - 4ac}}$

Can coefficient $a$ be zero?

No. If $a = 0$, the $x^2$ term vanishes and the equation becomes linear ($b x + c = 0$).

Standards & References

1. Stewart, J., Redlin, L., & Watson, S.: Algebra and Trigonometry. Comprehensive reference for quadratic equations and complex number roots.
2. Press, W. H., et al.: Numerical Recipes. Industry standard for numerically stable quadratic solvers preventing catastrophic floating-point cancellation.
3. NIST Digital Library of Mathematical Functions (DLMF): Standard reference for polynomial functions.