Quadratic equations are a fundamental concept in algebra. They represent relationships where one variable is squared, leading to curved graphs called parabolas. Understanding how to solve these equations is crucial for tackling more complex math problems.

The Quadratic Formula is a powerful tool for solving quadratic equations. It works for all quadratics, even when other methods fail. By analyzing the discriminant, we can determine the number and nature of solutions, giving us insights into the equation's behavior.

Solving Quadratic Equations

Application of Quadratic Formula

• Quadratic Formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ solves quadratic equations in standard form $ax^2 + bx + c = 0$
• Apply Quadratic Formula by identifying coefficients $a$, $b$, and $c$, substituting into formula, simplifying discriminant $b^2 - 4ac$, calculating two solutions by adding and subtracting square root term
• $\pm$ symbol represents two possible solutions: one with addition $(+)$ and one with subtraction $(-)$
• Example: Solve $2x^2 - 5x - 3 = 0$ using Quadratic Formula
  • $a = 2$, $b = -5$, $c = -3$
  • $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(2)(-3)}}{2(2)}$
  • $x = \frac{5 \pm \sqrt{25 + 24}}{4} = \frac{5 \pm \sqrt{49}}{4} = \frac{5 \pm 7}{4}$
  • Two solutions: $x = \frac{5 + 7}{4} = 3$ and $x = \frac{5 - 7}{4} = -\frac{1}{2}$
• These solutions are also called the roots of the quadratic equation

Interpretation of discriminant

• Discriminant $b^2 - 4ac$ determines nature and number of solutions for quadratic equation
  • Positive discriminant $(b^2 - 4ac > 0)$ indicates two distinct real solutions (parabola intersects x-axis at two points)
  • Zero discriminant $(b^2 - 4ac = 0)$ indicates one repeated real solution (parabola touches x-axis at one point)
  • Negative discriminant $(b^2 - 4ac < 0)$ indicates two complex solutions involving imaginary unit $i$ where $i^2 = -1$ (parabola does not intersect x-axis)
• Example: Determine number and nature of solutions for $3x^2 + 6x + 2 = 0$
  • $a = 3$, $b = 6$, $c = 2$
  • Discriminant $= 6^2 - 4(3)(2) = 36 - 24 = 12 > 0$
  • Two distinct real solutions exist

Choosing the Best Method

Methods for solving quadratics

• Factoring most efficient when quadratic has integer coefficients, leading coefficient $a$ is $\pm 1$, constant term $c$ factors into two integers with sum $b$
  • Example: $x^2 + 7x + 12 = 0$ factors as $(x + 3)(x + 4) = 0$
• Completing the square useful for rewriting equation in vertex form to identify axis of symmetry and vertex coordinates, deriving Quadratic Formula
  • Example: $x^2 + 6x + 5 = 0$ becomes $(x + 3)^2 - 4 = 0$ in vertex form
• Quadratic Formula most general method, applies when equation cannot be factored, coefficients are fractions or decimals, exact solution needed
  • Example: $2x^2 - \sqrt{3}x - \frac{1}{2} = 0$ solved using Quadratic Formula

Graphical Representation and Analysis

• Quadratic equations represent parabolas when graphed in the coordinate plane
• The solutions (roots) of a quadratic equation correspond to the x-intercepts of its graph
• Graphing a quadratic function can provide visual insight into the nature of its solutions
• Quadratic equations are a specific type of polynomial equation of degree 2