Problem 51 - Algebra and Trigonometry

Question

Find all real solutions of the equation. $$ 3 x^{2}+2 x+2=0 $$

Step-by-Step Solution

Answer

The equation has no real solutions.

Step 1: Identify the Quadratic Equation

The given equation is a quadratic equation, specifically in the form \( ax^2 + bx + c = 0 \). Here, \( a = 3 \), \( b = 2 \), and \( c = 2 \). We will find the roots of this equation.

Step 2: Determine the Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). We calculate it for our equation:\[ \Delta = 2^2 - 4 \times 3 \times 2 = 4 - 24 = -20 \].

Step 3: Analyze the Discriminant Result

Since the discriminant \( \Delta = -20 \) is less than zero, it indicates that the quadratic equation has no real solutions. Negative discriminants mean the roots are complex and not real.

Step 4: Conclude No Real Solutions

As the discriminant is negative, there are no intersections with the x-axis and therefore no real number solutions to the equation \( 3x^2 + 2x + 2 = 0 \).

Key Concepts

Understanding the Discriminant

The discriminant is a key part of the quadratic formula and helps us understand the nature of the roots of a quadratic equation. If you're dealing with a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant, represented by \( \Delta \), is calculated using the formula:

• \( \Delta = b^2 - 4ac \)

The value of the discriminant informs us about the type of roots that the equation will have:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If \( \Delta < 0 \), the equation has complex roots and no real solutions.

In our exercise, \( \Delta \) was found to be -20. This negative value confirms that the quadratic equation has complex roots, leading us to the conclusion that there are no real solutions.

Exploring Complex Roots

When the discriminant of a quadratic equation is less than zero, it indicates that the roots are complex numbers. Complex roots come in conjugate pairs, meaning they contain imaginary parts, which are symmetrical around the real axis. For example, if you solve a quadratic equation and find complex roots, they will typically be expressed as:

• \( x_1 = p + qi \)
• \( x_2 = p - qi \)

Here, \( p \) is the real part and \( q \) is the imaginary coefficient multiplied by \( i \), the imaginary unit defined by \( i^2 = -1 \).
In our specific problem, the discriminant value was -20, implying it's not possible to have real solutions. Instead, the roots can be calculated using the quadratic formula, yielding complex numbers. The quadratic formula is:

• \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \)

With this information, we establish that despite having no real solutions, the equation \( 3x^2 + 2x + 2 = 0 \) resolves to complex roots.

Identifying Real Solutions

A real solution to a quadratic equation occurs when the equation equals zero at real number values of \( x \). Here's how to identify whether real solutions exist:

• Calculate the discriminant, \( \Delta = b^2 - 4ac \).
• If \( \Delta \geq 0 \), the equation potentially has real solutions.
• These solutions could either be distinct (two different values) if \( \Delta > 0 \), or a single repeated root if \( \Delta = 0 \).

In situations such as the exercise: the discriminant \( \Delta = -20 \) results in no real solutions because \( \Delta < 0 \). Consequently, the equation does not meet the condition for having real numbers as solutions; rather, it leads us to complex roots. This scenario shows that sometimes analysis of the discriminant is enough to determine the impossibility of finding real solutions.