Problem 40 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}+6 x=-8 $$

Step-by-Step Solution

Answer

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

Step 1: Move Constant to Right Hand Side

Rearrange the equation so that the constant term -8 is alone on the right side of the equation: \(x^2 + 6x + 8 = 0\).

Step 2: Complete the Square

Modify the equation so the left-hand side of it is a perfect square. This can be done by adding the square of half the coefficient of x to both sides of the equation. Half of 6 is 3, and 3 squared is 9, so add 9 to both sides of the equation: \((x^2 + 6x + 9) = -8 + 9\).

Step 3: Simplify Equation

Simplify both sides of the equation. The left side should form a perfect square, and the right side is -8 + 9 which equals 1: \((x + 3)^2 = 1\).

Step 4: Find the Roots of the Equation

Take the square root of both sides to solve for x. Remember to consider both the positive and negative roots when dealing with squares: \(x + 3 = \pm \sqrt{1}\).

Step 5: Solve for X

Solving for x gives you the solution to the equation: \(x = -3 \pm 1\). This results in two solutions for the given problem: x = -2 and x = -4.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental algebraic expressions that feature an unknown variable, usually denoted as 'x', raised to the second power. Their general form is given by \(ax^2 + bx + c = 0\), where 'a', 'b', and 'c' are coefficients with 'a' being non-zero.

One commonly used method to solve quadratic equations is 'completing the square', which enables the transformation of a quadratic equation into a perfect square trinomial. Through this technique, finding the variable's value becomes a matter of algebraic maneuvers involving addition or subtraction, multiplication, and finding square roots. The equation presented, \(x^2 + 6x = -8\), is an example of such a problem where this method can successfully be applied for finding the solutions of the equation.

Completing the square is especially valuable when the quadratic does not factorize neatly or when using other methods such as the quadratic formula is not preferable due to its complex nature. By understanding this process, students can build a strong foundation for solving a wide range of algebraic problems.

Perfect Square Trinomials

A perfect square trinomial is an algebraic expression created by squaring a binomial. It takes the form \( (ax+b)^2 = a^2x^2 + 2abx + b^2 \), where the first and last terms are perfect squares, and the middle term is twice the product of 'a' and 'b'.

In the context of completing the square for solving quadratic equations, transforming a quadratic into a perfect square trinomial is the key step. It entails manipulating the equation so that one side becomes a perfect square trinomial, making it easier to solve.

To correctly transform a quadratic expression into a perfect square, one must identify the value to be added and subtracted, which is determined by taking half the coefficient of 'x' and squaring it. Incorporating this into the original equation allows you to neatly convert it into the form \( (x + m)^2 \), where 'm' is the value just calculated, simplifying the process of finding the roots of the equation.

Square Root Method

The square root method is used in solving quadratic equations once a perfect square trinomial is formed. After completing the square, the equation is in the form of \( (x + m)^2 = n \), where 'm' and 'n' are numbers.

To resolve the equation, the square root of both sides is taken to eliminate the square on one side, leading to \( x + m = \pm \sqrt{n} \). The '\pm' indicates that both the positive and negative square roots must be considered, as either could provide a valid solution to the original quadratic.

This method is particularly intuitive because, once the perfect square trinomial is established, applying the square root to both sides of the equation straightforwardly exposes the value of 'x'. As such, it's an excellent tool for simplifying the complexity of quadratic equations. However, it's important to remind students to check all potential solutions in the original equation to ensure they are valid.