To solve the quadratic equation $2x^2 + 3x - 2 = 0$ by completing the square, follow these steps:

Step 1: Move the constant to the other side

Start with the equation: $2x^2 + 3x - 2 = 0$ Move the constant $-2$ to the other side of the equation: $2x^2 + 3x = 2$

Step 2: Factor out the leading coefficient from the quadratic and linear terms

Factor out the $2$ from the left-hand side: $2(x^2 + \frac{3}{2}x) = 2$

Step 3: Complete the square

To complete the square, take half of the coefficient of $x$ (which is $\frac{3}{2}$), square it, and add it inside the parentheses. The number to add is: $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$ Add $\frac{9}{16}$ inside the parentheses, and also subtract it outside the parentheses to balance the equation: $2\left(x^2 + \frac{3}{2}x + \frac{9}{16}\right) - 2\left(\frac{9}{16}\right) = 2$

Step 4: Simplify the equation

Now, rewrite the expression inside the parentheses as a perfect square and simplify: $2\left(x + \frac{3}{4}\right)^2 - \frac{18}{16} = 2$ Simplify the fraction: $2\left(x + \frac{3}{4}\right)^2 - \frac{9}{8} = 2$

Step 5: Move the constant to the other side

Add $\frac{9}{8}$ to both sides: $2\left(x + \frac{3}{4}\right)^2 = 2 + \frac{9}{8}$ Combine the constants: $2\left(x + \frac{3}{4}\right)^2 = \frac{16}{8} + \frac{9}{8} = \frac{25}{8}$

Step 6: Solve for $x$

Divide both sides by 2: $\left(x + \frac{3}{4}\right)^2 = \frac{25}{16}$ Take the square root of both sides: $x + \frac{3}{4} = \pm \frac{5}{4}$

Step 7: Isolate $x$

Now solve for $x$ by subtracting $\frac{3}{4}$ from both sides: $x = -\frac{3}{4} \pm \frac{5}{4}$

This gives two solutions: $x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$ $x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$

Final Answer:

The solution set is: $x = \frac{1}{2}, -2$

Would you like more details or have any questions?

Related Questions:

1. How do you complete the square for equations with leading coefficients different from 1?
2. What is the geometric interpretation of completing the square?
3. How do you solve quadratic equations using the quadratic formula?
4. What are other methods to solve quadratic equations?
5. How do you check the solutions of a quadratic equation?

Tip:

Always verify your solutions by substituting them back into the original equation!