use complete the square to solve the equation 2x^2-3x-1=0

To solve the quadratic equation \( 2x^2 - 3x - 1 = 0 \) by completing the square, follow these steps:

### Step 1: Divide by the leading coefficient
We first make the coefficient of \( x^2 \) equal to 1 by dividing the entire equation by 2:

\[
x^2 - \frac{3}{2}x - \frac{1}{2} = 0
\]

### Step 2: Move the constant term to the other side
Move the constant term to the right-hand side of the equation:

\[
x^2 - \frac{3}{2}x = \frac{1}{2}
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \( x \), which is \( -\frac{3}{2} \), and square it:

\[
\left( \frac{-3/2}{2} \right)^2 = \left( \frac{-3}{4} \right)^2 = \frac{9}{16}
\]

Now, add this value to both sides of the equation:

\[
x^2 - \frac{3}{2}x + \frac{9}{16} = \frac{1}{2} + \frac{9}{16}
\]

### Step 4: Simplify the right-hand side
First, convert \( \frac{1}{2} \) to have a denominator of 16:

\[
\frac{1}{2} = \frac{8}{16}
\]

Now add the fractions:

\[
\frac{8}{16} + \frac{9}{16} = \frac{17}{16}
\]

Thus, the equation becomes:

\[
\left( x - \frac{3}{4} \right)^2 = \frac{17}{16}
\]

### Step 5: Solve for \( x \)
Now take the square root of both sides:

\[
x - \frac{3}{4} = \pm \sqrt{\frac{17}{16}} = \pm \frac{\sqrt{17}}{4}
\]

### Step 6: Isolate \( x \)
Finally, solve for \( x \):

\[
x = \frac{3}{4} \pm \frac{\sqrt{17}}{4}
\]

This simplifies to:

\[
x = \frac{3 \pm \sqrt{17}}{4}
\]

### Final Answer:
The solutions are:

\[
x = \frac{3 + \sqrt{17}}{4} \quad \text{or} \quad x = \frac{3 - \sqrt{17}}{4}
\]

Would you like a detailed explanation of any step or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you complete the square for equations with negative \(x^2\) terms?
2. How can you apply the quadratic formula to solve the same equation?
3. How do you solve quadratic equations with irrational coefficients?
4. What are the graphical interpretations of the roots of a quadratic equation?
5. How does completing the square help derive the quadratic formula?

**Tip:** Completing the square is also useful in converting a quadratic equation into vertex form, making it easier to graph.

Learn how to solve the quadratic equation 2x^2 - 3x - 1 = 0 using the method of completing the square. This step-by-step guide covers key algebraic concepts and explains the completing the square formula, ideal for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation