Problem 48 - College Algebra Essentials

Question

Solve each equation in Exercises \(47-64\) by completing the square. $$ x^{2}+6 x=-8 $$

Step-by-Step Solution

Answer

The solutions to the given equation \(x^{2}+6 x=-8\) are \(x=-2\) and \(x=-4\).

Step 1: Create a Perfect Square Trinomial

To start with, rewrite the equation to one side as: \(x^{2}+6 x+8=0\). Move the constant term 8 to the other side, this gives \(x^{2}+6 x = -8\). Next, take half of the coefficient of \(x\) (that is, 6/2 = 3), square it to get 9, and add that 9 on both sides of the equation. The left side of the equation then becomes a perfect square trinomial: \(x^{2}+6 x +9 = -8 +9\)

Step 2: Write the Left Side of the Equation as a Square

The perfect square trinomial on the left side can be written as a square: \((x+3)^{2}=1\). This is obtained by factoring the trinomial \(x^{2} +6x +9\), which gives \((x+3)\) as a factor, hence \((x+3)^2\).

Step 3: Solve for \(x\)

Finding the square root of both sides of this equation gives us two possible solutions: \(x+3=\sqrt{1}\) or \(x+3=-\sqrt{1}\). This simplifies to \(x+3=1\) or \(x+3=-1\). So, the solutions to the equation are \(x=1-3=-2\) and \(x=-1-3=-4\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These second-degree polynomials are notable because they graph as parabolas, offering a visual representation of their solutions or roots.
The standard form of a quadratic equation aids in categorizing and solving the equation by various methods such as factoring, using the quadratic formula, or completing the square.
Each method serves a unique purpose, with completing the square providing a structured approach to reformatting the equation into a perfect square trinomial, thereby making it easier to solve.

Forming a Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that results from squaring a binomial. For example, \((x + a)^2 = x^2 + 2ax + a^2\) yields a perfect square trinomial. Recognizing and forming perfect square trinomials is a vital step in solving quadratic equations by completing the square.
To create a perfect square trinomial from an equation, isolate the terms involving \(x\) and determine half of the linear coefficient (term involving \(x\)), then square it. This new value is the constant term needed to complete the square.

• Example: Given \(x^2 + 6x\), half of the coefficient of \(x\) is \(3\). Squaring \(3\), we get \(9\). Adding \(9\) to both sides of the equation converts \(x^2 + 6x = -8\) to the perfect square trinomial \(x^2 + 6x + 9\).

Factoring: Unveiling the Perfect Square

Once a perfect square trinomial is formed, factoring becomes a simple process. Factoring involves expressing a polynomial as a product of its factors. In the context of a perfect square trinomial, it simplifies into the square of a binomial.
For instance, the perfect square trinomial \(x^2 + 6x + 9\) factors neatly into \((x + 3)^2\). This factorization simplifies solving the equation.

• To achieve this: Begin with the trinomial and rewrite it as \((x + a)^2\), where \(a\) is the halved coefficient of \(x\) originally squared to form the perfect square trinomial.

By expressing it as a binomial squared, you directly solve the equation by taking the square root of both sides. This step uncovers potential roots of the original quadratic equation, offering a straightforward path to finding solutions.