Completing the Square Using the Quadratic Formula

When dealing with quadratic equations, one of the most effective methods to solve them is completing the square. This process transforms a quadratic equation into a perfect square trinomial, which can then be solved more easily. This article will guide you through the steps of completing the square using the quadratic formula, providing clear examples to illustrate the process.

Introduction to Quadratic Equations and the Quadratic Formula

The general form of a quadratic equation is given by ax^2 bx c 0, where a, b, and c are constants, and a ≠ 0. The quadratic formula, which can be directly applied to solve for x, is as follows:

x_{1,2} frac{-b pm sqrt{b^2 - 4ac}}{2a}

This formula provides the values of x that satisfy the equation, and generally, there are two such values, though sometimes they can be complex or imaginary.

Example 1: Solving x^2 - 4x - 12 0

Let's solve the quadratic equation x^2 - 4x - 12 0 using the quadratic formula:

1. Identify the coefficients: a 1, b -4, c -12.

2. Apply the quadratic formula:

x_{1,2} frac{-(-4) pm sqrt{(-4)^2 - 4 cdot 1 cdot (-12)}}{2 cdot 1}

3. Simplify the expression inside the square root:

x_{1,2} frac{4 pm sqrt{16 48}}{2}

4. Calculate the square root:

x_{1,2} frac{4 pm sqrt{64}}{2}

5. Simplify further:

x_{1,2} frac{4 pm 8}{2}

6. Solve for x:

x_1 frac{4 8}{2} 6

x_2 frac{4 - 8}{2} -2

Thus, the solutions are x_1 6 and x_2 -2.

Example 2: Solving x^2 - 3x - 12 0

Now, consider the equation x^2 - 3x - 12 0 to demonstrate a slightly trickier case:

1. Identify the coefficients: a 1, b -3, c -12.

2. Apply the quadratic formula:

x_{1,2} frac{-(-3) pm sqrt{(-3)^2 - 4 cdot 1 cdot (-12)}}{2 cdot 1}

3. Simplify the expression inside the square root:

x_{1,2} frac{3 pm sqrt{9 48}}{2}

4. Calculate the square root:

x_{1,2} frac{3 pm sqrt{57}}{2}

5. Simplify further:

Since sqrt{57} is not a perfect square, the roots will be imaginary.

Thus, the solutions are:

x_{1,2} frac{3}{2} pm frac{i}{2} sqrt{57}

Completing the Square

Another method to solve quadratic equations is by completing the square. This technique transforms a uncompleted square trinomial into a perfect square form, which simplifies the process of finding the roots.

Step-by-Step Guide to Completing the Square

Rearrange the quadratic equation so that the constant term is on the right side. For example, for ax^2 bx c 0, this becomes ax^2 bx -c. Divide every term by the coefficient of x^2, so the equation has the form x^2 frac{b}{a}x -frac{c}{a}. To complete the square, add and subtract left(frac{b}{2a}right)^2 on the left side of the equation. This results in:

x^2 frac{b}{a}x left(frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2

The left side of the equation now forms a perfect square trinomial:

left(x frac{b}{2a}right)^2 -frac{c}{a} left(frac{b}{2a}right)^2

Solve for x:

x frac{b}{2a} pm sqrt{-frac{c}{a} left(frac{b}{2a}right)^2}

Finally, isolate x:

x -frac{b}{2a} pm sqrt{-frac{c}{a} left(frac{b}{2a}right)^2}

Conclusion

In conclusion, completing the square and using the quadratic formula are both powerful methods for solving quadratic equations. While the quadratic formula provides a straightforward algebraic approach, completing the square offers a geometric and conceptual insight into why the formula works.

Key Terms

Quadratic Formula: A formula used to solve quadratic equations, which is given by x_{1,2} frac{-b pm sqrt{b^2 - 4ac}}{2a}.

Completing the Square: A method of transforming a square trinomial into a perfect square form, allowing for easier solution of quadratic equations.

These techniques are invaluable in various fields, including physics, engineering, and economics, where quadratic equations often arise.