Quadratic equations are a cornerstone of algebra, allowing us to solve problems involving squared terms. The quadratic formula is a powerful tool that can crack any quadratic equation, no matter how tricky it looks.

Understanding the discriminant helps us predict the nature of a quadratic equation's solutions before we even solve it. This insight is crucial for tackling more complex problems and interpreting real-world scenarios modeled by quadratic equations.

Solving Quadratic Equations

Application of quadratic formula

• Solves quadratic equations in the form $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants and $a \neq 0$
• Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • $a$, $b$, and $c$ are coefficients of the quadratic equation
  • $\pm$ symbol indicates two possible solutions
• Steps to apply quadratic formula:
  1. Identify values of $a$, $b$, and $c$ in given quadratic equation
  2. Substitute values into quadratic formula
  3. Simplify expression under square root (discriminant)
  4. Calculate two possible solutions by adding and subtracting square root term
• Example: Solve $2x^2 + 7x - 4 = 0$
  • $a = 2$, $b = 7$, $c = -4$
  • $x = \frac{-7 \pm \sqrt{7^2 - 4(2)(-4)}}{2(2)}$
  • $x = \frac{-7 \pm \sqrt{49 + 32}}{4} = \frac{-7 \pm \sqrt{81}}{4} = \frac{-7 \pm 9}{4}$
  • $x_1 = \frac{-7 + 9}{4} = \frac{1}{2}$ and $x_2 = \frac{-7 - 9}{4} = -4$

Interpretation of discriminant

• Discriminant is expression under square root in quadratic formula: $b^2 - 4ac$
• Value of discriminant determines nature and number of solutions for quadratic equation
  • Positive discriminant ($b^2 - 4ac > 0$): two distinct real solutions
  • Zero discriminant ($b^2 - 4ac = 0$): one repeated real solution
  • Negative discriminant ($b^2 - 4ac < 0$): no real solutions (two complex solutions)
• Example: Determine nature and number of solutions for $3x^2 - 5x + 2 = 0$
  • $a = 3$, $b = -5$, $c = 2$
  • Discriminant = $(-5)^2 - 4(3)(2) = 25 - 24 = 1$
  • Positive discriminant, so equation has two distinct real solutions

Methods for quadratic equations

• Factoring most efficient when:
  • Quadratic equation has integer coefficients
  • Leading coefficient ($a$) is 1 or easily factored out
  • Product of $a$ and $c$ is relatively small
• Completing the square most efficient when:
  • Quadratic equation not easily factored
  • Leading coefficient ($a$) is 1 or easily factored out
  • Equation needs transformation into vertex form ($y = a(x - h)^2 + k$)
• Quadratic formula most efficient when:
  • Quadratic equation not easily factored
  • Leading coefficient ($a$) not 1 and not easily factored out
  • Equation has complex ($a + bi$) or irrational ($\sqrt{2}$) solutions
• General approach: try factoring first, then completing the square if factoring not possible or efficient, finally use quadratic formula if other methods not suitable

Quadratic Equations and Their Graphical Representation

• Quadratic equations in standard form ($ax^2 + bx + c = 0$) represent parabolas when graphed
• Solutions to quadratic equations (roots) correspond to x-intercepts of the parabola
• The imaginary unit $i$ is used when the discriminant is negative, resulting in complex roots
• The quadratic formula helps find these roots, whether real or complex