Question

Use the quadratic formula to solve the equation. $$-\frac{2}{3} x^{2}-3 x+1=0$$

Short answer

The solutions to the equation \(-\frac{2}{3} x^{2}-3 x+1=0\) using the quadratic formula are \(x=\frac{3 + \sqrt{27/3}}{-4/3}\) and \(x=\frac{3 - \sqrt{27/3}}{-4/3}\).

Step-by-step solution

Identify Coefficients

First, recognize the given equation \(-\frac{2}{3} x^{2}-3 x+1=0\) is in the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). In this equation, \(a = -\frac{2}{3}\), \(b = -3\), and \(c = 1\).

Apply the Quadratic Formula

Substitute the coefficients \(a\), \(b\), and \(c\) into the quadratic formula \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). This gives \(\frac{-(-3) \pm \sqrt{(-3)^2-4(-2/3)(1)}}{2(-2/3)}\).

Solve under the square root

Calculate the term under the square root. This gives \(\frac{3 \pm \sqrt{9 - (4*-2/3)*1}}{-4/3}\).

Simplify the Expression

Further simplifying the expression gives \(\frac{3 \pm \sqrt{9+8/3}}{-4/3\). Continue by simplifying under the root which gives \(\frac{3 \pm \sqrt{27/3}}{-4/3}\). Consequentially, the final solutions are \(x=\frac{3 + \sqrt{27/3}}{-4/3}\) and \(x=\frac{3 - \sqrt{27/3}}{-4/3}\).