Solve the quadratic equation by completing the square.

2 x squared plus 3 x minus 52x2+3x−5equals=0

To solve the quadratic equation \( 2x^2 + 3x - 5 = 0 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
First, we rewrite the equation as:
\[
2x^2 + 3x = 5
\]

### Step 2: Divide the equation by the coefficient of \( x^2 \)
To complete the square, we need the coefficient of \( x^2 \) to be 1, so divide the entire equation by 2:
\[
x^2 + \frac{3}{2}x = \frac{5}{2}
\]

### Step 3: Complete the square
Now, take half of the coefficient of \( x \), which is \( \frac{3}{2} \), and square it. 
\[
\left(\frac{3}{4}\right)^2 = \frac{9}{16}
\]
Add \( \frac{9}{16} \) to both sides of the equation:
\[
x^2 + \frac{3}{2}x + \frac{9}{16} = \frac{5}{2} + \frac{9}{16}
\]

### Step 4: Simplify the right-hand side
To simplify the right-hand side, express \( \frac{5}{2} \) as \( \frac{40}{16} \), then add:
\[
\frac{40}{16} + \frac{9}{16} = \frac{49}{16}
\]
So, the equation becomes:
\[
\left(x + \frac{3}{4}\right)^2 = \frac{49}{16}
\]

### Step 5: Solve for \( x \)
Take the square root of both sides:
\[
x + \frac{3}{4} = \pm \frac{7}{4}
\]

### Step 6: Solve for \( x \)
Now, solve for \( x \) by subtracting \( \frac{3}{4} \) from both sides:
\[
x = -\frac{3}{4} \pm \frac{7}{4}
\]
This gives two solutions:
\[
x = \frac{-3 + 7}{4} = 1 \quad \text{or} \quad x = \frac{-3 - 7}{4} = -\frac{5}{2}
\]

### Final Solution:
The solutions are:
\[
x = 1 \quad \text{or} \quad x = -\frac{5}{2}
\]

Do you want more details or have any questions about this method?

Here are 5 questions that expand on this topic:
1. What is the difference between completing the square and factoring?
2. Can completing the square be used for any quadratic equation?
3. How can the method of completing the square be used to derive the quadratic formula?
4. What is the geometric interpretation of completing the square?
5. How does completing the square help in solving equations with complex numbers?

**Tip:** Completing the square is particularly useful for converting a quadratic equation into vertex form, which can be helpful for graphing parabolas.

Solve the quadratic equation by completing the square: 2x^2 + 3x - 5 = 0

Learn how to solve the quadratic equation 2x^2 + 3x - 5 using the completing the square method. This solution breaks down the process step by step, covering important algebraic concepts and formulas. Suitable for Grades 8-10 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation