2x² + x - 28 = 0 Solve by completing the square 

To solve the quadratic equation \(2x^2 + x - 28 = 0\) by completing the square, follow these steps:

### Step 1: Divide the entire equation by the coefficient of \(x^2\)
The coefficient of \(x^2\) is 2, so divide every term by 2:

\[
x^2 + \frac{1}{2}x - 14 = 0
\]

### Step 2: Move the constant term to the right side of the equation
Add 14 to both sides to isolate the quadratic and linear terms:

\[
x^2 + \frac{1}{2}x = 14
\]

### Step 3: Complete the square
To complete the square, take half of the coefficient of \(x\) (which is \(\frac{1}{2}\)), square it, and add it to both sides of the equation. 

Half of \(\frac{1}{2}\) is \(\frac{1}{4}\), and squaring it gives \(\left(\frac{1}{4}\right)^2 = \frac{1}{16}\).

Add \(\frac{1}{16}\) to both sides:

\[
x^2 + \frac{1}{2}x + \frac{1}{16} = 14 + \frac{1}{16}
\]

### Step 4: Simplify both sides
The left side of the equation is now a perfect square trinomial:

\[
\left(x + \frac{1}{4}\right)^2 = \frac{224}{16} + \frac{1}{16} = \frac{225}{16}
\]

### Step 5: Take the square root of both sides
Take the square root of both sides to solve for \(x\):

\[
x + \frac{1}{4} = \pm \sqrt{\frac{225}{16}}
\]

Simplifying further:

\[
x + \frac{1}{4} = \pm \frac{15}{4}
\]

### Step 6: Solve for \(x\)
Finally, solve for \(x\) by subtracting \(\frac{1}{4}\) from both sides:

\[
x = -\frac{1}{4} \pm \frac{15}{4}
\]

This gives two solutions:

\[
x = \frac{14}{4} = \frac{7}{2} = 3.5
\]

\[
x = -\frac{16}{4} = -4
\]

### Final Answer:
The solutions to the equation are \(x = 3.5\) and \(x = -4\).

Would you like more details on any of the steps, or do you have any questions?

### Relative Questions:
1. How do you factor a quadratic equation instead of completing the square?
2. What is the quadratic formula, and how can it be used to solve any quadratic equation?
3. Can completing the square be used for all quadratic equations?
4. How does completing the square relate to the vertex form of a quadratic function?
5. What are the advantages and disadvantages of solving quadratic equations by completing the square versus other methods?

### Tip:
When completing the square, always ensure that the coefficient of \(x^2\) is 1 by dividing the entire equation by the leading coefficient before proceeding.

2x² + x - 28 = 0 Solve by completing the square

Learn how to solve the quadratic equation 2x² + x - 28 by completing the square with this detailed, step-by-step solution. Explore key algebraic concepts, such as completing the square, and apply mathematical formulas to find the solutions x = 3.5 and x = -4. Suitable for students in Grades 9-10.

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