Problem 73 - Elementary Algebra

Question

\(n^{2}+5 n+6=0\)

Step-by-Step Solution

Answer

The solutions are \(n = -2\) and \(n = -3\).

Step 1 - Identify the Quadratic Equation

Recognize that the given equation \(n^{2} + 5n + 6 = 0\) is a quadratic equation, where \(a=1\), \(b=5\), and \(c=6\).

Step 2 - Use the Quadratic Formula

The quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) can be applied to find the solutions. Substitute \(a=1\), \(b=5\), and \(c=6\) into the formula.

Step 3 - Calculate the Discriminant

Calculate the discriminant \(D = b^2 - 4ac\). For this equation, \(D = 5^2 - 4(1)(6) = 25 - 24 = 1\).

Step 4 - Solve for n

Substitute the discriminant \(D\) back into the quadratic formula: \(n = \frac{-5 \pm \sqrt{1}}{2(1)}\). This simplifies to \(n = \frac{-5 \pm 1}{2}\).

Step 5 - Find the Two Solutions

Separate the equation into two parts: \(n = \frac{-5 + 1}{2} = -2\) and \(n = \frac{-5 - 1}{2} = -3\).

Key Concepts

Discriminant

The discriminant is a key part of the quadratic equation, playing a significant role in determining the types and number of solutions. In any quadratic equation of the form \[ ax^2 + bx + c = 0 \], the discriminant (D) can be calculated using the formula: \[ D = b^2 - 4ac \] .
Here's why the discriminant is important:

• If D > 0, the quadratic equation has two distinct real solutions.
• If D = 0, the equation has exactly one real solution.
• If D < 0, the equation has no real solutions; instead, it has two complex solutions.

In our exercise, we calculated the discriminant for the equation \[n^2 + 5n + 6 = 0\] using \[a = 1\], \[b = 5\] , and \[c = 6\] . This gave us: \[D = 5^2 - 4(1)(6) = 25 - 24 = 1\] . Since \[D > 0\] , we know there are two distinct real solutions.

Quadratic Formula

The quadratic formula is a universal way to solve any quadratic equation. It's based on the coefficients of the equation \[ax^2 + bx + c = 0\] . The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula uses the discriminant \( b^2 - 4ac \) under the square root to find solutions. Here's how it works in our exercise:
1. We identified the coefficients: \[a = 1\], \[b = 5\], \[c = 6\]
2. We substitute these values into the quadratic formula: \[n = \frac{-5 \pm \sqrt{1}}{2(1)}\].
3. Simplifying further, we get: \[n = \frac{-5 \pm 1}{2}\].This step-by-step application allows us to solve any quadratic equation methodically.

Solving quadratic equations

Solving quadratic equations involves finding the values of the variable that make the quadratic equation true. Here is the detailed method to solve our example \[n^2 + 5n + 6 = 0\] using the quadratic formula:
1. **Write down the equation:** We start with \[n^2 + 5n + 6 = 0\].
2. **Identify the coefficients:** Recognize \[a = 1\], \[b = 5\] and \[c = 6\].
3. **Compute the discriminant:** Calculate \[D = b^2 - 4ac\], which gives us \[D = 25 - 24 = 1\].
4. **Apply the quadratic formula:** Substitute \[a\], \[b\] and \[D\] into the formula: \[n = \frac{-5 \pm \sqrt{1}}{2(1)}\].
5. **Simplify the equation:** Break it down into two parts:

• \[n = \frac{-5 + 1}{2} = -2\]
• \[n = \frac{-5 - 1}{2} = -3\]

By following these steps, we solved the quadratic equation and found the solutions n = -2 and n = -3. This method is not only effective but can also be generalized to solve any quadratic equation you encounter.