Completing the Square Study Notes

Completing the Square: Detailed Study Notes

Concept Overview

• Completing the square is a method used in algebra to solve quadratic equations.
• A quadratic equation is any equation that can be rearranged in the standard form: $ax^2 + bx + c = 0$ where:
  • $a$ is the coefficient of $x^2$
  • $b$ is the coefficient of $x$
  • $c$ is the constant term.

Steps to Complete the Square

1. Start with the quadratic equation:
   • Make sure it is in the standard form.
2. Isolate the constant term ($ c $):
   • Rearrange the equation to bring the constant term on the right side:
     $ax^2 + bx = -c$
3. Factor out the coefficient of $x^2$ ($ a $) from the left side:
   • If $a <br />\neq 1$, factor it out:
     $a(x^2 + \frac{b}{a}x) = -c$
4. Complete the square:
   • Take half of the coefficient of $x$, square it, and add it to both sides:
     $a \left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a} \right)^2 \right) = -c + a\left(\frac{b}{2a}\right)^2$
5. Rewrite the left side as a squared term:
   • This can be written as:
     $a\left(x + \frac{b}{2a}\right)^2 = -c + \frac{b^2}{4a}$
6. Solve for $x$ by taking the square root:
   • Isolate the squared term and solve:
     $x + \frac{b}{2a} = \pm \sqrt{-c + \frac{b^2}{4a}}$
   • Thus,
     $x = -\frac{b}{2a} \pm \sqrt{-c + \frac{b^2}{4a}}$

Example Problems

1. Solve using the completing the square:

   • Given Equation:
     $3x^2 - 2x - 6 = 0$
   • Step 1: Rearranging yields:
     $3x^2 - 2x = 6$
   • Step 2: Factor out the 3:
     $3(x^2 - \frac{2}{3}x) = 6$
   • Step 3: Completing the square involves calculating $\left(\frac{-2/3}{2}\right)^2 = \left(\frac{-1/3}{2}\right)^2 = \frac{1}{36}$
   • Step 4: Add and subtract this value inside the brackets:
     $3\left(x^2 - \frac{2}{3}x + \frac{1}{36} - \frac{1}{36}\right) = 6$
   • Step 5: Simplifying gives:
     $3\left((x - \frac{1}{3})^2 - \frac{1}{36}\right) = 6$
   • Step 6: Introduce the constant correction:
     $3(x - \frac{1}{3})^2 - \frac{1}{12} = 6$
   • Step 7: Solving yields:
     $3(x - \frac{1}{3})^2 = 6 + \frac{1}{12}$, followed by
     $(x - \frac{1}{3})^2 = \frac{73}{36}$
   • Step 8: Combine and solve for $x$.

2. Second Example:

   • Given Equation:
     $4x^2 + 3x - 7 = 0$
   • Proceed similarly as above, following outlined steps.

3. Third Example:

   • Given Equation:
     $2x^2 - x - 12 = 0$
   • Again, follow similar procedures to isolate and complete the square.

Additional Notes

• The concept not only aids in solving equations but also helps in understanding the vertex form of a quadratic function:
  $y = a(x - h)^2 + k$
  where (h, k) is the vertex.
• Applications include physics (projectile motion), economics (profit maximization), and any field dealing with parabolic trajectories.