Understanding the Quadratic Formula and Its Applications

Overview of the Quadratic Formula

• The quadratic formula is used to find the roots of a quadratic function consistently and accurately.

Steps to Utilize the Quadratic Formula

1. Standard Form: Ensure the quadratic equation is in standard form, which is
   $ax^2 + bx + c = 0$.

2. Identify Coefficients: Determine the values of a, b, and c from the equation.

3. Substitution: Substitute these values into the quadratic formula:
   $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

4. Simplify: Simplify the expression to find the roots.

Example 1: Solving $x^2 - 7x - 30 = 0$

• Identify the coefficients:

  • a = 1, b = -7, c = -30

• Plug into the quadratic formula:
  $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-30)}}{2(1)}$

• Calculate discriminant:

  • $(-7)^2 = 49$

  • $-4(1)(-30) = 120$

  • Therefore, $49 + 120 = 169$.

• Complete the formula:
  $x = \frac{7 \pm \sqrt{169}}{2}$.

• Calculate the square root:

  • $ext{Since }\sqrt{169} = 13$, the equation becomes:
    $x = \frac{7 \pm 13}{2}$.

• Solutions:

  • When adding:
    $x = \frac{20}{2} = 10$

  • When subtracting:
    $x = \frac{-6}{2} = -3.$

Example 2: Solving $x^2 - 3x = -6$

1. Rearrange the equation:

   • $x^2 - 3x + 6 = 0$.

2. Identify the coefficients:

   • a = 1, b = -3, c = 6.

3. Plug into the quadratic formula:
   $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(6)}}{2(1)}$.

4. Calculate discriminant:

   • $(-3)^2 = 9$

   • $-4(1)(6) = -24$

   • Therefore, $9 - 24 = -15$.

5. Complete the formula:
   $x = \frac{3 \pm \sqrt{-15}}{2}$.

6. Solve for imaginary roots:

   • $x = \frac{3 \pm i\sqrt{15}}{2}$.

   • Note: Cannot combine like terms easily since the solution involves an imaginary component.

Key Takeaways

• The quadratic formula is a reliable method for finding roots of quadratic equations.

• Careful attention to detail during each step is crucial for accuracy.

• The nature of the roots (real or imaginary) can be determined by evaluating the discriminant, $b^2 - 4ac$.