Explore Westonci.ca, the premier Q&A site that helps you find precise answers to your questions, no matter the topic. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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.
### 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.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.