Problem 42
Question
\(n^{2}+2 n+1=0\)
Step-by-Step Solution
Verified Answer
The solution is \( n = -1 \).
1Step 1 - Identify the equation type
This equation is a quadratic equation in the form of \( ax^{2} + bx + c = 0 \), where \(a=1\), \(b=2\), and \(c=1\).
2Step 2 - Apply the quadratic formula
The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \). Substituting \(a=1\), \(b=2\), and \(c=1\), we get: \[ n = \frac{-2 \pm \sqrt{2^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
3Step 3 - Simplify under the square root
Calculate the term under the square root: \( 2^{2} - 4 \cdot 1 \cdot 1 = 4 - 4 = 0 \). Thus, the equation simplifies to: \[ n = \frac{-2 \pm \sqrt{0}}{2} \]
4Step 4 - Compute the solutions
Since \( \sqrt{0} = 0 \), the equation simplifies to: \[ n = \frac{-2}{2} = -1 \]. There is a single solution: \( n = -1 \).
Key Concepts
Understanding the Quadratic FormulaSimplifying Radicals in the Quadratic FormulaIdentifying a Single Solution for Quadratic Equations
Understanding the Quadratic Formula
The quadratic equation is a staple in algebra and appears in the form: \( ax^{2} + bx + c = 0 \). Each coefficient and constant plays a critical role in finding the solution. To solve this, we use the quadratic formula, made specifically to handle any standard quadratic equation.
The standard quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) are the coefficients from the equation \( ax^{2} + bx + c = 0 \).
Using this formula ensures that we can find the roots (or solutions) of the quadratic equation efficiently. For example, consider the quadratic equation from the exercise: \( n^{2} + 2n + 1 = 0 \). Here, \( a = 1 \), \( b = 2 \), and \( c = 1 \). Plug these values into the formula as follows: \[ n = \frac{-2 \pm \sqrt{2^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \].
This crucial step sets the foundation for solving the equation.
The standard quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] where \(a\), \(b\), and \(c\) are the coefficients from the equation \( ax^{2} + bx + c = 0 \).
Using this formula ensures that we can find the roots (or solutions) of the quadratic equation efficiently. For example, consider the quadratic equation from the exercise: \( n^{2} + 2n + 1 = 0 \). Here, \( a = 1 \), \( b = 2 \), and \( c = 1 \). Plug these values into the formula as follows: \[ n = \frac{-2 \pm \sqrt{2^{2} - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \].
This crucial step sets the foundation for solving the equation.
Simplifying Radicals in the Quadratic Formula
After substituting the coefficients into the quadratic formula, the next step involves simplifying the term under the square root, known as the discriminant: \( b^{2} - 4ac \).
Simplifying this component is essential as it influences the nature and number of solutions. For the example \( n^{2} + 2n + 1 = 0 \), calculate: \( b^{2} - 4ac = 2^{2} - 4 \cdot 1 \cdot 1 = 4 - 4 = 0 \).
The discriminant in this example simplifies to 0, which we recognize as having a significant impact.
Next, we simplify the square root of the discriminant (\( \sqrt{0} \)). Since \( \sqrt{0} = 0 \), the quadratic equation further simplifies to: \[ n = \frac{-2 \pm 0}{2} \] which simplifies just to: \[ n = \frac{-2}{2} = -1 \].
This process of simplifying makes it clear how the discriminant helps determine the exact nature of the roots.
Simplifying this component is essential as it influences the nature and number of solutions. For the example \( n^{2} + 2n + 1 = 0 \), calculate: \( b^{2} - 4ac = 2^{2} - 4 \cdot 1 \cdot 1 = 4 - 4 = 0 \).
The discriminant in this example simplifies to 0, which we recognize as having a significant impact.
Next, we simplify the square root of the discriminant (\( \sqrt{0} \)). Since \( \sqrt{0} = 0 \), the quadratic equation further simplifies to: \[ n = \frac{-2 \pm 0}{2} \] which simplifies just to: \[ n = \frac{-2}{2} = -1 \].
This process of simplifying makes it clear how the discriminant helps determine the exact nature of the roots.
Identifying a Single Solution for Quadratic Equations
A critical outcome of solving quadratic equations is understanding the discriminant's role in determining the number of solutions.
The discriminant (\( b^{2} - 4ac \)) essentially tells us how many roots we should expect:
As calculated, the solution is \[ n = \frac{-2}{2} = -1 \],which is our single root. This scenario is a perfect example of finding a single solution for a quadratic equation and demonstrates its simplicity when the discriminant is zero.
The discriminant (\( b^{2} - 4ac \)) essentially tells us how many roots we should expect:
- If \( b^{2} - 4ac > 0 \), there are two distinct solutions.
- If \( b^{2} - 4ac = 0 \), there is exactly one single solution.
- If \( b^{2} - 4ac < 0 \), there are no real solutions, only complex ones.
As calculated, the solution is \[ n = \frac{-2}{2} = -1 \],which is our single root. This scenario is a perfect example of finding a single solution for a quadratic equation and demonstrates its simplicity when the discriminant is zero.