What is Completing the Square?

Completing the square is a method used in algebra to solve quadratic equations, convert quadratic expressions into a different form, and analyze the properties of parabolas. It’s a handy technique that transforms a quadratic equation into a perfect square trinomial plus or minus a constant. Let’s dive into how it works.

Steps to Complete the Square

  1. Start with a Quadratic Equation
    Consider a standard quadratic equation of the form:
    $ax^2 + bx + c = 0$

  1. Move the Constant Term
    First, move the constant term to the other side of the equation:
    $ax^2 + bx = -c$

  1. Divide by the Coefficient of $x^2$
    If the coefficient of $x^2$ is not 1, divide the entire equation by $a$:
    $x^2 + frac{b}{a}x = -frac{c}{a}$

  1. Add and Subtract a Number
    To complete the square, add and subtract the square of half the coefficient of $x$ inside the equation. The coefficient of $x$ is $frac{b}{a}$, so half of it is $frac{b}{2a}$. Squaring this gives $left(frac{b}{2a}right)^2$:
    $x^2 + frac{b}{a}x + left(frac{b}{2a}right)^2 = -frac{c}{a} + left(frac{b}{2a}right)^2$

  1. Write as a Perfect Square Trinomial
    The left side of the equation is now a perfect square trinomial, which can be written as:
    $left(x + frac{b}{2a}right)^2 = -frac{c}{a} + left(frac{b}{2a}right)^2$

  1. Solve for $x$
    Finally, solve for $x$ by taking the square root of both sides and isolating $x$:
    $x + frac{b}{2a} = pmsqrt{-frac{c}{a} + left(frac{b}{2a}right)^2}$
    $x = -frac{b}{2a} pmsqrt{left(frac{b}{2a}right)^2 – frac{c}{a}}$

Example

Let’s go through an example to see this method in action. Solve the quadratic equation $x^2 + 6x + 5 = 0$ using completing the square.

  1. Move the Constant Term
    $x^2 + 6x = -5$

  1. Add and Subtract a Number
    The coefficient of $x$ is 6, so half of it is 3. Squaring this gives 9. Add and subtract 9 inside the equation:
    $x^2 + 6x + 9 = -5 + 9$

  1. Write as a Perfect Square Trinomial
    $left(x + 3right)^2 = 4$

  1. Solve for $x$
    $x + 3 = pm2$
    $x = -3 pm 2$

    So, the solutions are $x = -1$ and $x = -5$

Applications of Completing the Square

Completing the square is not just for solving equations. It’s also used in:

  • Graphing Parabolas: Helps to find the vertex form of a quadratic equation.
  • Integration: Useful in calculus for integrating certain rational functions.
  • Physics: Appears in equations of motion and other physical formulas.

Conclusion

Completing the square is a versatile and powerful algebraic technique. By transforming a quadratic equation into a perfect square trinomial, it simplifies solving and analyzing the equation, making it easier to understand and work with.

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