Problem 14 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}+6 x=-8\)

Step-by-Step Solution

Answer

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

Step 1: Rearrange the Equation

Move -8 from the right side to the left side of equation to make it on standard form: \(x^{2}+6x+8=0\)

Step 2: Complete the Square

To complete the square, take half the coefficient of 'x', square it and then add and subtract it inside the parenthesis: \((x^2 + 6x + 9 - 1) = 0\(x + 3)^2 -1 = 0\). This now turned into perfect square trinomial.

Step 3: Solving for 'x'

Step 1: Move -1 to the right side of equation: \((x+3)^2=1\).Step 2: Take the square root of both sides of the equation: \(x+3=\pm \sqrt{1}\).Step 3: Subtract 3 from both sides of equation to isolate 'x': \(x= -3 \pm 1\).Step 4: The roots of the equation are \(x=-4\) (if you subtract 1) and \(x=-2\) (if you add 1).

Step 4: Verify the Solution

Substitute \(x=-4\) and \(x=-2\) into original equation to confirm the roots. Both should satisfy the equation.

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental concept in algebra, representing an equation of the second degree. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This structure signifies that the highest degree of the variable \(x\) is two. Quadratic equations model various physical phenomena and processes, from the shape of a projectile's path to calculations in economics.

When understanding quadratic equations, it’s crucial to know that they can have two, one, or no real solutions. These are the x-values, also known as the roots, which satisfy the equation when replaced for \(x\). The roots can be found using several methods, one of which is completing the square, known for its effectiveness in making equations easier to solve or graph.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation. Various methods exist for solving them, such as factoring, using the quadratic formula, and completing the square. Each method has its advantages and ideal scenarios for use. In the problem we've reviewed, completing the square is used to find the roots.

Completing the square transforms a quadratic equation into a form that makes it straightforward to solve. Here's a simple outline of how it is done:

• Step 1: Align the equation in the form \(x^2 + bx = c\), if necessary.
• Step 2: Find \(\frac{b}{2}\), square it, and add and subtract this square on the left side of the equation.
• Step 3: Rewrite the left side as a squared binomial, and simplify the equation.
• Step 4: Solve for \(x\) by taking square roots and isolating \(x\).

These steps simplify the process, ensuring clear and manageable progression to the solution.

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic expression that takes the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). When expanded, it results in a quadratic expression that can be rewritten easily in its binomial squared form.

In the context of completing the square, recognizing and forming a perfect square trinomial is key. For instance, in the equation transformation step of the problem \(x^2 + 6x + 9\), by adding \(9\), we formed a perfect square trinomial \((x + 3)^2\). This made it straightforward to solve the equation by extracting the square root.

Understanding perfect square trinomials allows you to convert quadratic equations into a form where extracting square roots and solving for \(x\) becomes direct. It helps visualize how terms in a quadratic expression relate to one another, providing powerful intuition into solving these equations efficiently and accurately.