Understanding the Completion of the Square: Intermediate Step for x2 - 66x - 15 0

In this comprehensive article, we will walk through the process of completing the square for the quadratic equation x2 - 66x - 15 0. We will break down the steps involved in rearranging and solving the equation, with a specific focus on the intermediate step that transforms the equation into the form x - a2 b.

Introduction to Quadratics and Completing the Square

Quadratic equations are fundamental in algebra, typically written in the standard form ax2 bx c 0. The process of completing the square is a method used to solve these equations by reformulating them into a perfect square trinomial. This method is particularly useful for finding the roots of the equation, revealing its solutions in a clearer manner.

Step-by-Step Guide to Completing the Square

The given equation x2 - 66x - 15 0 can be solved by completing the square. Here are the detailed steps in this process:

Rearranging the Equation

First, the equation is rearranged to bring all terms to one side, setting the equation equal to zero:

x2 - 14x - 66 - 15 0

Simplifying this gives:

x2 - 14x - 51 0

Completing the Square

The next step in completing the square involves moving the constant term to the right side of the equation:

x2 - 14x -51

To complete the square, take half of the coefficient of x, which is -14, square it, and add it to both sides of the equation. Half of -14 is -7, and squaring this gives 49 (since (-7)2 49):

x2 - 14x 49 -51 49

This simplifies to:

x2 - 14x 49 -2

The left side of the equation is now a perfect square:

(x - 7)2 -2

Intermediate Step

The intermediate step in the form x - a2 b is: x - 72 -2.

Complex Solutions and Roots

If we consider the complex number system, the equation (x - 7)2 -2 can be solved by taking the square roots of both sides. Extracting the roots, we get:

x - 7 /- sqrt(-2)

Since the square root of a negative number involves the imaginary unit i, we can write:

x - 7 /- sqrt(2)i

Therefore, the roots are:

x 7 /- sqrt(2)i

Rewriting the roots:

x 7 - sqrt(2)i and x 7 sqrt(2)i

Alternative Methods for Solving Quadratic Equations

In addition to the completing the square method, a quadratic equation can also be solved using the quadratic formula, which is expressed as:

x (-b ± sqrt(b2 - 4ac)) / (2a)

Applying this to the equation x2 - 66x - 15 0, where a 1, b -66, and c -15, we get:

x (66 ± sqrt((-66)2 - 4(1)(-15))) / (2(1))

Simplifying the expression under the square root:

x 66 ± sqrt(4356 60)

x 66 ± sqrt(4416)

This can be further simplified to:

x 66 ± sqrt(49 * 90)

x 66 ± 7 * sqrt(90)

Since sqrt(90) 3 * sqrt(10), we get:

x 66 ± 21 * sqrt(10)

Congruently, the roots can be expressed as:

x 66 21 * sqrt(10) and x 66 - 21 * sqrt(10)

This gives us the exact solutions for the quadratic equation in terms of real and complex numbers.

By understanding the process of completing the square and applying alternative methods to solve quadratic equations, you can gain a deeper appreciation for the algebraic techniques that are foundational in mathematics.