Certainly! Let's solve the quadratic equation [tex]\(2x^2 + x - 4 = 0\)[/tex] by completing the square, step-by-step.
### Step 1: Normalize the quadratic coefficient
First, we want to make the coefficient of [tex]\(x^2\)[/tex] equal to 1. We do this by dividing the entire equation by 2.
[tex]\[
\frac{2x^2 + x - 4}{2} = 0
\][/tex]
[tex]\[
x^2 + \frac{1}{2}x - 2 = 0
\][/tex]
### Step 2: Move the constant term to the other side
Next, we move the constant term [tex]\(-2\)[/tex] to the right side of the equation.
[tex]\[
x^2 + \frac{1}{2}x = 2
\][/tex]
### Step 3: Complete the square
To complete the square, we need to add and subtract the same value on the left-hand side such that it forms a perfect square trinomial. We take half the coefficient of [tex]\(x\)[/tex] (which is [tex]\(\frac{1}{2}\)[/tex]), square it, and add it to both sides.
[tex]\[
\left( \frac{1}{2} \cdot \frac{1}{2} \right)^2 = \left( \frac{1}{4} \right)^2 = \frac{1}{16}
\][/tex]
Now we add [tex]\(\frac{1}{16}\)[/tex] to both sides:
[tex]\[
x^2 + \frac{1}{2}x + \frac{1}{16} = 2 + \frac{1}{16}
\][/tex]
### Step 4: Rewrite the left side as a binomial squared
The left-hand side now forms a perfect square trinomial:
[tex]\[
\left( x + \frac{1}{4} \right)^2 = 2 + \frac{1}{16}
\][/tex]
### Step 5: Simplify the right side
Combine the terms on the right side:
[tex]\[
2 + \frac{1}{16} = \frac{32}{16} + \frac{1}{16} = \frac{33}{16}
\][/tex]
So we have:
[tex]\[
\left( x + \frac{1}{4} \right)^2 = \frac{33}{16}
\][/tex]
### Final Result
Let's summarize our completed square form:
[tex]\[
\left( x + \frac{1}{4} \right)^2 = \frac{33}{16}
\][/tex]
This is the result after solving the quadratic equation by completing the square.
