Problem 129 - College Algebra

Question

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 \(\sqrt{2} x^{2}+3 x-2 \sqrt{2}=0\) are \(x = \frac{-3 + \sqrt{17}}{2}\) and \(x = \frac{-3 - \sqrt{17}}{2}\)

Step 1: Identify the coefficients

The equation given is \(\sqrt{2} x^{2}+3 x-2 \sqrt{2}=0\). Here, \(a = \sqrt{2}\), \(b = 3\), and \(c = -2\sqrt{2}\)

Step 2: Apply the quadratic formula

We can solve this quadratic equation by using the quadratic formula: \(x = \frac{-b \pm \sqrt{{b^2} - 4ac}}{2a}\). Replacing \(a\), \(b\) and \(c\) with their respective values from the given equation-\(x = \frac{-3 \pm \sqrt{{3^2} - 4*(\sqrt{2})*(-2\sqrt{2})}}{2*\sqrt{2}}\)

Step 3: Simplify the Expression

Upon simplification, the solution to the equation is \(x = \frac{-3 \pm \sqrt{9 + 8}}{2\sqrt{2}}\) = \(\frac{-3 \pm \sqrt{17}}{2\sqrt{2}}\). Now use rationalizing factor to get rid of square root value in denominator. Ultimately the solutions are \(x = \frac{-3 + \sqrt{17}}{2}\) or \(x = \frac{-3 - \sqrt{17}}{2}\).