Sure, let's solve the quadratic equation [tex]\(x^2 + 4x - 1 = 0\)[/tex] step-by-step by completing the square.
### Step 1: Completing the Square
Given the equation:
[tex]\[x^2 + 4x - 1 = 0\][/tex]
1. First, move the constant term to the other side:
[tex]\[x^2 + 4x = 1\][/tex]
2. To complete the square, we need to add and subtract the square of half the coefficient of [tex]\(x\)[/tex]. The coefficient of [tex]\(x\)[/tex] is 4, so half of it is 2, and squaring it gives us 4. So, we add and subtract 4 inside the equation:
[tex]\[x^2 + 4x + 4 - 4 = 1\][/tex]
3. Write the equation in the form of a perfect square trinomial:
[tex]\[(x + 2)^2 - 4 = 1\][/tex]
4. Now, isolate the perfect square term:
[tex]\[(x + 2)^2 = 5\][/tex]
### Step 2: Solving the Equation
Now we have the equation [tex]\((x + 2)^2 = 5\)[/tex]. To solve for [tex]\(x\)[/tex], take the square root of both sides:
[tex]\[ x + 2 = \pm \sqrt{5} \][/tex]
This gives us two solutions:
[tex]\[ x + 2 = \sqrt{5} \quad \text{or} \quad x + 2 = -\sqrt{5} \][/tex]
Next, solve for [tex]\(x\)[/tex]:
1. For [tex]\( x + 2 = \sqrt{5} \)[/tex]:
[tex]\[ x = \sqrt{5} - 2 \][/tex]
2. For [tex]\( x + 2 = -\sqrt{5} \)[/tex]:
[tex]\[ x = -\sqrt{5} - 2 \][/tex]
So, the solutions are:
[tex]\[ x = -2 + \sqrt{5} \][/tex]
[tex]\[ x = -2 - \sqrt{5} \][/tex]
### Choosing the Correct Answer
Given the options:
- (A) [tex]\( x = 2 \pm 5 \)[/tex]
- (B) [tex]\( x = -2 \pm 5 \)[/tex]
- (C) [tex]\( x = 2 \pm \sqrt{5} \)[/tex]
- (D) [tex]\( x = -2 \pm \sqrt{5} \)[/tex]
The correct answer is:
[tex]\[ \text{(D)} \quad x = -2 \pm \sqrt{5} \][/tex]
So the solutions to the equation [tex]\(x^2 + 4x - 1 = 0\)[/tex] are [tex]\(x = -2 + \sqrt{5}\)[/tex] and [tex]\(x = -2 - \sqrt{5}\)[/tex].
