16) \( \begin{array}{l}3 x^{2}+5 x-2=0< \\ x^{2}+\frac{5 x}{3}=\frac{12}{3} \cdot\left[\frac{\frac{5}{3}}{2}\right]^{2}=\begin{array}{l}x^{2}+\frac{5 x}{3}-\frac{2}{3}=0 \\ \downarrow \\ x^{2}+\frac{5 x}{3}+\frac{25}{36}=\frac{2}{3}+\frac{25}{36}\end{array}\end{array} \)

Let's break down the problem step by step. ### Step 1: Solve the first equation The first equation is: \[ 3x^2 + 5x - 2 = 0 \] We will solve this quadratic equation using the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \(a = 3\), \(b = 5\), and \(c = -2\). ### Step 2: Calculate the discriminant First, we need to calculate the discriminant \(D\): \[ D = b^2 - 4ac = 5^2 - 4 \cdot 3 \cdot (-2) \] ### Step 3: Solve for \(x\) Now, we will use the quadratic formula to find the values of \(x\). ### Step 4: Solve the second equation The second equation is: \[ x^2 + \frac{5x}{3} - \frac{2}{3} = 0 \] We will also solve this quadratic equation using the quadratic formula. Here, \(a = 1\), \(b = \frac{5}{3}\), and \(c = -\frac{2}{3}\). ### Step 5: Calculate the discriminant for the second equation We will calculate the discriminant \(D\) for the second equation: \[ D = \left(\frac{5}{3}\right)^2 - 4 \cdot 1 \cdot \left(-\frac{2}{3}\right) \] ### Step 6: Solve for \(x\) in the second equation Now, we will use the quadratic formula to find the values of \(x\) for the second equation. Let's perform these calculations. Solve the equation by following steps: - step0: Solve for \(x\): \(x^{2}+\frac{5x}{3}-\frac{2}{3}=0\) - step1: Multiply both sides of the equation by LCD: \(\left(x^{2}+\frac{5x}{3}-\frac{2}{3}\right)\times 3=0\times 3\) - step2: Simplify the equation: \(3x^{2}+5x-2=0\) - step3: Factor the expression: \(\left(x+2\right)\left(3x-1\right)=0\) - step4: Separate into possible cases: \(\begin{align}&x+2=0\\&3x-1=0\end{align}\) - step5: Solve the equation: \(\begin{align}&x=-2\\&x=\frac{1}{3}\end{align}\) - step6: Rewrite: \(x_{1}=-2,x_{2}=\frac{1}{3}\) Solve the equation \( 3x^2 + 5x - 2 = 0 \). Solve the quadratic equation by following steps: - step0: Solve by factoring: \(3x^{2}+5x-2=0\) - step1: Factor the expression: \(\left(x+2\right)\left(3x-1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&3x-1=0\\&x+2=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=\frac{1}{3}\\&x=-2\end{align}\) - step4: Rewrite: \(x_{1}=-2,x_{2}=\frac{1}{3}\) ### Step 1: Solutions to the first equation For the equation: \[ 3x^2 + 5x - 2 = 0 \] The solutions are: \[ x_1 = -2, \quad x_2 = \frac{1}{3} \quad \text{(or } 0.3\text{)} \] ### Step 2: Solutions to the second equation For the equation: \[ x^2 + \frac{5x}{3} - \frac{2}{3} = 0 \] The solutions are the same: \[ x_1 = -2, \quad x_2 = \frac{1}{3} \quad \text{(or } 0.3\text{)} \] ### Summary of Results Both equations yield the same solutions: - \(x_1 = -2\) - \(x_2 = \frac{1}{3}\) (or \(0.3\)) If you have any further questions or need additional assistance, feel free to ask!