Quadratic Equations: Completing the Square

Completing the square is a technique used to solve quadratic equations, which are equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The process involves transforming the quadratic equation into a perfect square trinomial, making it easier to find the solutions or roots of the equation. Below is a detailed explanation of how to complete the square and solve quadratic equations using this method.

1. Start with the Quadratic Equation:
   Begin with a quadratic equation in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are known constants.

   Example: $2x^2 + 8x – 6 = 0$

2. Divide by the Leading Coefficient (if necessary):
   If the leading coefficient $a$ is not 1, divide the entire equation by $a$. This step ensures that the coefficient of $x^2$ is 1.

   Continuing the example: $2x^2 + 8x – 6 = 0$ becomes $x^2 + 4x – 3 = 0$

3. Move the Constant Term to the Other Side:
   Move the constant term $c$ to the right side of the equation, leaving space to complete the square on the left side.

   Continuing the example: $x^2 + 4x = 3$

4. Create a Perfect Square Trinomial:
   To complete the square, add and subtract $\left(\frac{b}{2}\right)^2$ inside the parentheses on the left side of the equation. Here, $b$ is the coefficient of $x$.

   In our example, $b = 4$, so $\left(\frac{b}{2}\right)^2 = \left(\frac{4}{2}\right)^2 = 4$. Add and subtract 4 inside the parentheses:

   $x^2 + 4x + 4 – 4 = 3$

5. Factor the Perfect Square Trinomial:
   The left side of the equation should now be a perfect square trinomial, which can be factored into $(x + \frac{b}{2})^2$.

   Continuing the example: $(x + 2)^2 – 4 = 3$

6. Isolate the Square Term:
   Move the constant term on the left side to the right side of the equation to isolate the square term.

   Continuing the example: $(x + 2)^2 = 7$

7. Take the Square Root:
   Take the square root of both sides of the equation to solve for $x$. Remember to consider both the positive and negative square roots.

   $x + 2 = \pm \sqrt{7}$

8. Solve for $x$:
   Finally, solve for $x$ by subtracting 2 from both sides of the equation.

   $x = -2 \pm \sqrt{7}$

Thus, the solutions to the quadratic equation $2x^2 + 8x – 6 = 0$ are $x = -2 + \sqrt{7}$ and $x = -2 – \sqrt{7}$.

Completing the square is a fundamental method in algebra that allows you to find the solutions to quadratic equations without relying on factoring or the quadratic formula directly. It’s particularly useful when factoring or using the quadratic formula is challenging or not straightforward.

Completing the square is a fundamental algebraic technique used to solve quadratic equations and is also employed in various mathematical applications, such as deriving the quadratic formula, solving optimization problems, and graphing quadratic functions. Here, we will delve deeper into the process of completing the square and explore its broader significance in mathematics.

Understanding the Process of Completing the Square

1. Foundation of Quadratic Equations:
   Quadratic equations are polynomial equations of degree two, typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $x$ is the variable. The solutions to quadratic equations are known as the roots or zeros of the equation.

2. Objective of Completing the Square:
   The primary goal of completing the square is to rewrite a quadratic equation in the form $(x + p)^2 = q$, where $p$ and $q$ are constants. This form makes it easier to solve for $x$ and understand the behavior of the quadratic function geometrically.

3. Steps in Completing the Square:

   • Begin with a quadratic equation in standard form: $ax^2 + bx + c = 0$.
   • If $a \neq 1$, divide the entire equation by $a$ to make the coefficient of $x^2$ equal to 1.
   • Move the constant term $c$ to the other side of the equation.
   • Add and subtract $\left(\frac{b}{2}\right)^2$ inside the parentheses to create a perfect square trinomial.
   • Factor the perfect square trinomial on the left side.
   • Isolate the square term on one side of the equation by moving the constant term to the other side.
   • Take the square root of both sides to solve for $x$.

4. Applications and Significance:

   • Derivation of Quadratic Formula: Completing the square is an essential step in deriving the quadratic formula, which provides a direct method for finding the roots of any quadratic equation.
   • Graphical Interpretation: Completing the square helps in graphing quadratic functions by transforming them into vertex form, $y = a(x – h)^2 + k$, where $(h, k)$ represents the vertex of the parabola.
   • Optimization Problems: In optimization problems, completing the square is used to find the maximum or minimum value of a quadratic function, often representing quantities such as area, volume, or profit.
   • Complex Numbers: Completing the square is also used in dealing with complex numbers, especially when solving quadratic equations with non-real solutions.

5. Example Application:
   Consider the quadratic equation $3x^2 – 12x + 5 = 0$.

   • Divide by 3 to make the coefficient of $x^2$ equal to 1: $x^2 – 4x + \frac{5}{3} = 0$.
   • Move the constant term to the other side: $x^2 – 4x = -\frac{5}{3}$.
   • Add and subtract $\left(\frac{4}{2}\right)^2 = 4$ inside the parentheses: $x^2 – 4x + 4 – 4 = -\frac{5}{3}$.
   • Factor the perfect square trinomial: $(x – 2)^2 = \frac{7}{3}$.
   • Isolate the square term: $(x – 2)^2 = \frac{7}{3}$.
   • Take the square root: $x – 2 = \pm \sqrt{\frac{7}{3}}$.
   • Solve for $x$: $x = 2 \pm \sqrt{\frac{7}{3}}$.

6. General Formula for Completing the Square:
   The general process of completing the square can be summarized using the following steps:

   • Start with $ax^2 + bx + c = 0$.
   • Rewrite as $a(x^2 + \frac{b}{a}x) = -c$.
   • Add and subtract $\left(\frac{b}{2a}\right)^2$ inside the parentheses: $a\left(x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 – \left(\frac{b}{2a}\right)^2\right) = -c$.
   • Factor the perfect square trinomial: $a\left(\left(x + \frac{b}{2a}\right)^2 – \left(\frac{b}{2a}\right)^2\right) = -c$.
   • Isolate the square term and solve for $x$ as described earlier.

In conclusion, completing the square is a versatile method in algebra that facilitates the solution of quadratic equations and plays a crucial role in various mathematical contexts, including graphing, optimization, and complex numbers. Mastering this technique is essential for a deeper understanding of quadratic functions and their applications in mathematics and other disciplines.