Problem 46 - 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: Identify the nature of the equation

The given equation \(3 + 5z + z^2 = 0\) is a quadratic equation. It can be rewritten in the standard form as \(z^2 + 5z + 3 = 0\).

Step 2: Use the quadratic formula

For a quadratic equation of the form \(az^2 + bz + c = 0\), the solutions are given by the quadratic formula: \(z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, \(a = 1\), \(b = 5\), and \(c = 3\).

Step 3: Calculate the discriminant

The discriminant \(D\) is given by \(b^2 - 4ac\). Here, \(D = 5^2 - 4 \cdot 1 \cdot 3 = 25 - 12 = 13\). Since \(D > 0\), there are two distinct real solutions.

Step 4: Find the solutions

Substitute the values of \(a\), \(b\), \(c\), and \(D\) into the quadratic formula: \(z = \frac{-5 \pm \sqrt{13}}{2}\). Thus, the solutions are \(z = \frac{-5 + \sqrt{13}}{2}\) and \(z = \frac{-5 - \sqrt{13}}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions for quadratic equations. These equations typically take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The quadratic formula is expressed as:
\[z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
By substituting the values of \(a\), \(b\), and \(c\) from the given equation into this formula, you can calculate the solutions—or roots—of the quadratic equation.

• The term under the square root, \(b^2 - 4ac\), is called the discriminant.
• The symbol "±" indicates that the formula provides two solutions.

Remember, using the quadratic formula is a systematic way to find the roots, and it works even when the roots are not easily factorable integers.

Real Solutions

In the context of quadratic equations, real solutions refer to the values of the variable that satisfy the equation, and they are real numbers.
When working with the quadratic formula, the nature of the solutions (whether they are real or complex) is influenced by the discriminant:

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution (which is a repeated or double root).
• If the discriminant is negative, the solutions are complex and occur in conjugate pairs.

In this particular problem, since the discriminant \(D = 13\) is greater than zero, the quadratic equation \(3 + 5z + z^2 = 0\) has two distinct real solutions.

Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is the expression \(b^2 - 4ac\). It provides critical information about the roots of the equation without having to solve for them explicitly.
Calculating the discriminant:

• Helps determine the number of real solutions.
• Indicates whether the solutions are distinct or repeated.
• Identifies if the solutions are real or complex.

For the equation \(3 + 5z + z^2 = 0\), the discriminant is \(5^2 - 4 \times 1 \times 3 = 25 - 12 = 13\). This tells us that the equation has two distinct real solutions because \(D > 0\). Understanding the discriminant is crucial for quickly assessing the nature of a quadratic equation's solutions.

Nature of the Equation

The nature of a quadratic equation refers to the behavior and type of its solutions. This is primarily determined by the discriminant \(D = b^2 - 4ac\):

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), there is exactly one real root, which is repeated.
• If \(D < 0\), the roots are complex and are not real numbers.

For the equation \(z^2 + 5z + 3 = 0\), the discriminant is 13, meaning the nature of this equation is such that it has two distinct real solutions. Recognizing these characteristics helps predict and interpret the solutions you will get using the quadratic formula, demonstrating a deeper understanding of quadratic equations beyond just computation.