Sure, let's solve the equation [tex]\( x^2 + 4x + y^2 - 5 = 0 \)[/tex] step by step.
### Step 1: Understanding the Equation
We start with the equation:
[tex]\[ x^2 + 4x + y^2 - 5 = 0 \][/tex]
This is a standard form of a conic section, specifically a circle, ellipse, parabola, or hyperbola.
### Step 2: Grouping the Variables
Let's focus on grouping the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms:
[tex]\[ x^2 + 4x + y^2 - 5 = 0 \][/tex]
### Step 3: Completing the Square for x
To make it easier to solve for [tex]\( y \)[/tex], we can complete the square for the [tex]\( x \)[/tex]-terms:
[tex]\[ x^2 + 4x = (x + 2)^2 - 4 \][/tex]
So, substituting this back into the original equation, we get:
[tex]\[ (x + 2)^2 - 4 + y^2 - 5 = 0 \][/tex]
[tex]\[ (x + 2)^2 + y^2 - 9 = 0 \][/tex]
### Step 4: Isolating the [tex]\( y^2 \)[/tex] Term
Rearrange the equation to isolate the [tex]\( y^2 \)[/tex] term:
[tex]\[ (x + 2)^2 + y^2 = 9 \][/tex]
### Step 5: Solving for [tex]\( y \)[/tex]
Now isolate [tex]\( y^2 \)[/tex]:
[tex]\[ y^2 = 9 - (x + 2)^2 \][/tex]
### Step 6: Taking the Square Root
To solve for [tex]\( y \)[/tex], take the square root of both sides:
[tex]\[ y = \pm \sqrt{9 - (x + 2)^2} \][/tex]
### Step 7: Simplify the Solution
Express the solutions for [tex]\( y \)[/tex] more clearly:
Let [tex]\( (x + 2) = z \)[/tex], then the formula becomes:
[tex]\[ y = \pm \sqrt{9 - z^2} \][/tex]
Substitute [tex]\( z \)[/tex] back:
[tex]\[ y = \pm \sqrt{9 - (x + 2)^2} \][/tex]
### Step 8: Factorize the Inner Term
Notice that [tex]\( 9 - (x + 2)^2 \)[/tex] should be refactored:
[tex]\[ 9 - (x + 2)^2 = 9 - (x^2 + 4x + 4) \][/tex]
[tex]\[ = -x^2 - 4x + 5 = -1((x - 1)(x + 5))\][/tex]
Thus we have:
[tex]\[ y = \pm \sqrt{-(x - 1)(x + 5)}\][/tex]
### Final Answer
Hence, the solutions for the equation [tex]\( x^2 + 4x + y^2 - 5 = 0 \)[/tex] are:
[tex]\[ y = \pm \sqrt{-(x - 1)(x + 5)} \][/tex]
