Problem 129 - College Algebra Essentials

Question

In Exercises \(127-130,\) solve each equation by the method of your choice. $$ \sqrt{2} x^{2}+3 x-2 \sqrt{2}=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = \frac{-3\sqrt{2}}{4} + \frac{\sqrt{34}}{4}\) and \(x = \frac{-3\sqrt{2}}{4} - \frac{\sqrt{34}}{4}\).

Step 1: Identification of coefficients

Identify the coefficients \(a\), \(b\), and \(c\) from the given equation. In this equation, \(a = \sqrt{2}\), \(b = 3\), and \(c = -2 \sqrt{2}\).

Step 2: Application of the quadratic formula

Apply the quadratic formula \[x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}\] by substituting the values of \(a\), \(b\), and \(c\) into the formula. This gives \[x = \frac{-3 \pm \sqrt{3² - 4 * \sqrt{2} * -2 \sqrt{2}}}{2*\sqrt{2}}\]

Step 3: Simplification

Simplify the expression under the square root and the denominator. After simplifying, we get \[x = \frac{-3 \pm \sqrt{9 + 8}}{2\sqrt{2}}\], which further simplifies to \[x = \frac{-3 \pm \sqrt{17}}{2\sqrt{2}}\]

Step 4: Simplify further

To rationalize the denominator, multiply the numerator and the denominator by \(\sqrt{2}\), which gives \[x = \frac{-3\sqrt{2} \pm \sqrt{34}}{4}\]. Then divide each of the terms in the numerator by 4 to get two solutions, which is \[x = \frac{-3\sqrt{2}}{4} + \frac{\sqrt{34}}{4}\] or \[x = \frac{-3\sqrt{2}}{4} - \frac{\sqrt{34}}{4}\].

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool in algebra used to find the roots of any quadratic equation. A quadratic equation has the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The quadratic formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

To use the formula, you need to identify the coefficients of the equation, as done in this exercise. Note that the expression under the square root, \(b^2 - 4ac\), is called the discriminant. It plays an essential role in determining the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root (a repeated root).
• If \(b^2 - 4ac < 0\), there are two complex roots.

Knowing how to apply this formula can make solving quadratic equations much more straightforward, as it gives a direct path to the solutions.

Solving Equations

Solving equations is a critical skill in algebra, encompassing methods for finding unknown values. For quadratic equations, several methods are available:

• Factoring: If the equation can be factored easily, this method is often fast and efficient.
• Completing the Square: This method involves transforming the equation into a perfect square trinomial, helping to solve for the roots.
• Using the Quadratic Formula: Works for any quadratic and is particularly useful when other methods are cumbersome.

In the exercise, the quadratic formula was chosen due to its reliability in providing exact solutions, irrespective of the complexity of the coefficients.

Algebraic Solutions

Algebraic solutions provide exact values of the variables involved. Unlike numerical approximations, algebraic solutions show the precise symbolic representations of the roots. In the context of solving quadratic equations, utilizing algebraic methods like the quadratic formula ensures accuracy in results. This method involves several steps, starting from identifying coefficients, substituting them into the formula, simplifying the expression under the square root (discriminant), and further simplifying to get the final roots. The algebraic approach also includes:

• Simplification: Breaking down complex expressions to their simplest form.
• Rationalization: Converting an irrational denominator to a rational one, as seen in the simplification in the example solution.

Mastering these steps can serve well in solving various algebraic problems efficiently.

Coefficient Identification

Identifying coefficients is the first crucial step in solving any quadratic equation using the quadratic formula. The given equation is \(\sqrt{2} x^2 + 3x - 2 \sqrt{2} = 0\). Here, the coefficients are noted as follows:

• \(a\): The coefficient of \(x^2\), which is \(\sqrt{2}\).
• \(b\): The coefficient of \(x\), which is \(3\).
• \(c\): The constant term, \(-2\sqrt{2}\).

Understanding how to identify these components is paramount, as they are directly used in the quadratic formula to find the solutions. Misidentification can lead to incorrect solutions, which highlights their importance in solving quadratic equations effectively.