Problem 41 - Algebra and Trigonometry

Question

Find all real solutions of the equation. \(3+5 z+z^{2}=0\)

Step-by-Step Solution

Answer

The real solutions are \( z = \frac{-5 + \sqrt{13}}{2} \) and \( z = \frac{-5 - \sqrt{13}}{2} \).

Step 1: Recognize the Quadratic

The given equation is a quadratic equation in the form of \( z^2 + 5z + 3 = 0\). This means we need to use methods like factoring, completing the square, or the quadratic formula to find the solutions.

Step 2: Determine Discriminant

Calculate the discriminant of the quadratic equation using the formula \(b^2 - 4ac\). Here \(a = 1\), \(b = 5\), and \(c = 3\). The discriminant is \(5^2 - 4 \times 1 \times 3 = 25 - 12 = 13\).

Step 3: Analyze the Discriminant

Since the discriminant is 13, which is greater than zero, the quadratic equation has two distinct real solutions.

Step 4: Use the Quadratic Formula

Use the quadratic formula to find the solutions:\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values: \(a = 1\), \(b = 5\), and \(c = 3\) into the formula:\[ z = \frac{-5 \pm \sqrt{13}}{2} \]

Step 5: Calculate Solutions

Simplify the expressions to get the solutions. The solutions are:\[ z_1 = \frac{-5 + \sqrt{13}}{2} \] \[ z_2 = \frac{-5 - \sqrt{13}}{2} \]

Key Concepts

Understanding the Discriminant

When dealing with quadratic equations like \( z^2 + 5z + 3 = 0 \), the discriminant is a vital tool for identifying the nature of the roots. The discriminant is calculated with the formula \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).
In our example:

• \( a = 1 \)
• \( b = 5 \)
• \( c = 3 \)

So, the discriminant is \( 5^2 - 4 \times 1 \times 3 = 25 - 12 = 13\).
This number tells us about the number and type of solutions:

• If the discriminant is greater than zero, like in our case (13), there are two distinct real solutions.
• If it's equal to zero, there is exactly one real solution.
• If it's less than zero, the solutions are not real numbers (they are complex numbers).

Using the Quadratic Formula

The quadratic formula is a powerful equation that lets us find solutions to any quadratic equation, no matter whether it can be easily factored or not.
The formula is written as:\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]By applying it to our equation \( z^2 + 5z + 3 = 0 \), and using the discriminant we calculated (13), we substitute:

• \( a = 1 \)
• \( b = 5 \)
• \( c = 3 \)

This gives us:\[ z = \frac{-5 \pm \sqrt{13}}{2} \]
The symbols \( \pm \) mean there are two possible solutions, one with addition and the other with subtraction, which we’ll calculate next.

Finding Real Solutions

Since the discriminant is positive, we know there are two distinct real solutions for our quadratic equation.
The quadratic formula gives us two expressions due to the \( \pm \) sign:

• \( z_1 = \frac{-5 + \sqrt{13}}{2} \)
• \( z_2 = \frac{-5 - \sqrt{13}}{2} \)

These are the real solutions of the equation \( 3 + 5z + z^2 = 0 \).
These solutions can be further approximated to decimal values if needed, but in exact terms, they beautifully showcase how the quadratic formula works in providing precise answers.