Problem 35 - Algebra 1: Concepts and Skills

Question

Determine whether the equation has two solutions, one solution, or no real solution. (Lesson 9.7) $$ 6 x^{2}-x+5=0 $$

Step-by-Step Solution

Answer

The given equation has no real solutions.

Step 1: Identify coefficients

First, recognize the coefficients in the quadratic equation \( 6x^2 - x + 5 = 0 \). Here, \( a = 6 \), \( b = -1 \), and \( c =5 \). These coefficients are what's needed for the discriminant formula.

Step 2: Calculate the Discriminant

Now, calculate the discriminant using the formula \( b^{2} - 4ac \). Substituting the identified coefficients from Step 1 into this formula, we get \( (-1)^{2} - 4(6)(5) = 1 - 120 = -119 \).

Step 3: Determine the number of solutions

The discriminant is less than 0 (-119). Therefore, the equation has no real solutions. In other words, the parabola represented by this equation does not intersect the x-axis.

Key Concepts

Understanding the Discriminant

When solving quadratic equations, the discriminant provides valuable information about the nature of the solutions one might expect to find. Specifically, the discriminant is part of the quadratic formula, represented as the expression under the square root sign: \( b^2 - 4ac \).

For any quadratic equation in the standard form \( ax^2 + bx + c = 0 \), the discriminant can determine the number and type of solutions:

• A positive discriminant implies two distinct real solutions, meaning the parabola crosses the x-axis at two points.
• A discriminant of zero means there is exactly one real solution, indicating the parabola touches the x-axis at a single point, also known as the vertex.
• A negative discriminant, as in our exercise where we found it to be -119, indicates there are no real solutions. Instead, this leads us to explore complex numbers, as the solutions are non-real.

Understanding the value and sign of the discriminant is critical as it guides us toward the correct method for finding solutions and interpreting the graph of the quadratic function.

Applying the Quadratic Formula

The quadratic formula is a powerful tool that provides the solutions to any quadratic equation. It's stated as \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}} }}{{2a}} \), where 'a', 'b', and 'c' are the coefficients from the standard form of the quadratic equation mentioned earlier.

The beauty of the quadratic formula lies in its comprehensive nature; it can find real or complex solutions regardless of the discriminant's value. When the discriminant is negative, as in the given exercise, the square root of a negative number introduces us to complex numbers, which leads us to the next core concept. In cases where the discriminant is positive or zero, the quadratic formula simplifies to reveal real number solutions. Applying the quadratic formula is a straightforward process:

• Identify the coefficients from the quadratic equation.
• Substitute these coefficients into the formula.
• Simplify the expression to find your solutions.

It is a robust method that ensures no solution is left behind.

Complex Numbers and Non-Real Solutions

Complex numbers come into play when we encounter a negative discriminant. These numbers expand the real number system by including the square roots of negative numbers, often represented as \( i \), where \( i = \sqrt{{-1}} \).

In our exercise, because the discriminant was -119, we are dealing with the square root of a negative number, which cannot be simplified within the realm of real numbers. Instead, we express the solutions using complex numbers:

• The general form of a complex number is \( a + bi \), where 'a' is the real part and 'bi' is the imaginary part.
• In the context of a quadratic solution, the square root of the negative discriminant (in this exercise, \( \sqrt{{-119}} \)) would be expressed as \( i\sqrt{{119}} \).
• The complete solutions to the quadratic equation would then be written as two complex conjugates, relying on the \( \pm \) symbol from the quadratic formula.

Recognizing when to use complex numbers is crucial for accurately solving quadratics with no real solutions, as it allows us to fully explore the set of possible solutions.