Quadratic Equations

Quadratic Equations

1. Introduction to Quadratic Equations

1.1 Definition

A quadratic equation is a polynomial equation of degree 2 in the form:

where $a \neq 0$, $b$, and $c$ are constants, and $x$ is the variable.

1.2 Standard Form

The general quadratic equation is written as:

• $a$: Coefficient of $x^2$.
• $b$: Coefficient of $x$.
• $c$: Constant term.

1.3 Examples

1. $x^2 + 5x + 6 = 0$
2. $2x^2 - 3x + 1 = 0$

2. Solution of Quadratic Equations

2.1 Methods of Solving Quadratic Equations

1. Factoring

   • Split the middle term and factorize.
   • Example: Solve $x^2 + 5x + 6 = 0$
     • Factorize: $(x + 2)(x + 3) = 0$
     • Roots: $x = -2$, $x = -3$.

2. Completing the Square

   • Rewrite the equation to form a perfect square trinomial.
   • Example: Solve $x^2 + 6x + 5 = 0$
     • Rewrite: $x^2 + 6x + 9 = 4$
     • Simplify: $(x + 3)^2 = 4$
     • Roots: $x = -3 \pm 2$, i.e., $x = -1$ or $x = -5$.

3. Quadratic Formula

   • Use the formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
   • Example: Solve $2x^2 - 3x + 1 = 0$
     • $a = 2, b = -3, c = 1$
     • Substitute: $$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(1)}}{2(2)}$$ $$x = \frac{3 \pm \sqrt{9 - 8}}{4}$$ $$x = \frac{3 \pm 1}{4}$$ Roots: $x = 1$, $x = 0.5$.

4. Graphical Method

   • Plot $y = ax^2 + bx + c$.
   • The roots are the $x$-intercepts of the graph.

3. Forming a Quadratic Equation with Given Roots

3.1 Relationship Between Roots and Coefficients

For roots $p$ and $q$, the quadratic equation is:

• Sum of Roots: $p + q = -\frac{b}{a}$
• Product of Roots: $pq = \frac{c}{a}$

3.2 Example

Given Roots: $3$ and $-4$

• Sum: $p + q = 3 + (-4) = -1$
• Product: $pq = 3 \times (-4) = -12$
• Quadratic Equation: $$x^2 - (-1)x + (-12) = 0$$ $$x^2 + x - 12 = 0$$

4. Applications of Solutions of Quadratic Equations

4.1 Real-World Applications

1. Physics:

   • Calculating time of flight in projectile motion: $$h = ut + \frac{1}{2}gt^2$$ Rearranging gives a quadratic equation in $t$.

2. Engineering:

   • Finding stress and strain relationships in materials.

3. Economics:

   • Calculating maximum profit or revenue using parabolic cost or revenue functions.

4. Architecture:

   • Designing parabolic arches or bridges.

5. Business Optimization:

   • Solving equations for break-even points.

4.2 Example: Area of a Rectangle

• Problem: The length of a rectangle is 2 meters more than its width, and the area is 24 square meters. Find the dimensions.
  • Let the width = $x$.
  • Length = $x + 2$.
  • Area = $x(x + 2) = 24$.
  • Quadratic Equation: $x^2 + 2x - 24 = 0$.
  • Solving: $$x = \frac{-2 \pm \sqrt{2^2 - 4(1)(-24)}}{2(1)} = \frac{-2 \pm \sqrt{100}}{2}$$ $$x = 4 \, \text{(only positive root is valid)}.$$
  • Width = $4 \, \text{m}$, Length = $6 \, \text{m}$.

5. Common Misconceptions

1. Ignoring the Discriminant:

   • The discriminant ($b^2 - 4ac$) determines the nature of roots:
     • $b^2 - 4ac > 0$: Two real roots.
     • $b^2 - 4ac = 0$: One real root (repeated).
     • $b^2 - 4ac < 0$: Complex roots.

2. Confusing Roots with Coefficients:

   • Misinterpreting roots as coefficients leads to incorrect equations.

6. Summary

• Quadratic Equations are second-degree polynomial equations.
• Solution Methods: Factoring, completing the square, quadratic formula, and graphical analysis.
• Formation from Roots: Use sum and product relationships to derive equations.
• Applications: Widely used in physics, engineering, economics, and real-life optimization problems.

Key Formulae

1. Quadratic Formula:

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

2. Forming Equations:

   $$x^2 - (p + q)x + pq = 0$$

Use these principles to analyze and solve quadratic problems in various practical contexts!