Problem 35 - Elementary Algebra

Question

\(p^{2}+6 p+2=0\)

Step-by-Step Solution

Answer

The roots are \( p = -3 + \sqrt{7} \) and \( p = -3 - \sqrt{7} \)

Step 1: Identify the quadratic equation

The given quadratic equation is in the standard form: \(p^{2} + 6p + 2 = 0\)

Step 2: Calculate the discriminant

The discriminant for a quadratic equation \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). Here, \(a = 1\), \(b = 6\), and \(c = 2\). So, \( \text{Discriminant} = 6^2 - 4 \times 1 \times 2 = 36 - 8 = 28\)

Step 3: Use the quadratic formula

The roots of the quadratic equation can be found using the quadratic formula: \( p = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \). Substituting the values, we get: \( p = \frac{-6 \, \pm \, \sqrt{28}}{2} \)

Step 4: Simplify the roots

Simplify the roots: \[ p = \frac{-6 \, \pm \, \sqrt{28}}{2}\ = \frac{-6 \, \pm \, 2\sqrt{7}}{2} = -3 \, \pm \, \sqrt{7}\]

Step 5: State the final roots

The solutions to the quadratic equation are: \( p = -3 + \sqrt{7} \) and \( p = -3 - \sqrt{7} \)

Key Concepts

Discriminant

The discriminant is an important part of solving quadratic equations because it helps determine the nature of the roots. In a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant (\