Problem 48 - Algebra A Combined Function

Question

Use the discriminant to determine the number and types of solutions of each equation. $$ 5-4 x+12 x^{2}=0 $$

Step-by-Step Solution

Answer

The equation has two complex solutions, as the discriminant is negative.

Step 1: Write the equation in standard form

The given equation is:\[ 5 - 4x + 12x^2 = 0 \]We rewrite it in standard quadratic form as:\[ 12x^2 - 4x + 5 = 0 \]Here, the standard form is \( ax^2 + bx + c = 0 \) with \( a = 12 \), \( b = -4 \), and \( c = 5 \).

Step 2: Calculate the discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:\[ \Delta = b^2 - 4ac \]Substituting the values \( a = 12 \), \( b = -4 \), and \( c = 5 \):\[ \Delta = (-4)^2 - 4 \times 12 \times 5 \]\[ \Delta = 16 - 240 \]\[ \Delta = -224 \]

Step 3: Interpret the discriminant

The discriminant \( \Delta = -224 \) is negative. A negative discriminant indicates that the quadratic equation has two complex solutions (conjugate pairs), and they are not real.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that takes the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and the term \( x \) represents the unknown variable. Quadratic equations are unique because they have a degree of 2, meaning the highest exponent of \( x \) is 2. This degree is what typically results in the characteristic parabolic shape when graphed.
Quadratic equations can have different types of solutions:

• Real and distinct solutions
• Real and identical solutions (repeated roots)
• Complex (non-real) solutions

These solutions can help us understand the behavior of the quadratic equation in terms of roots and graph intersections with the x-axis.

Complex Solutions

Complex solutions arise in the context of quadratic equations when the discriminant is negative. The discriminant, represented by \( \Delta = b^2 - 4ac \), helps us determine the nature of the roots without actually solving the equation.
When \( \Delta < 0 \), the quadratic equation does not have any real solutions but instead has complex solutions, which occur in the form of conjugate pairs. A complex solution typically appears as:

• \( x = \frac{-b + i \sqrt{\left| \Delta \right|}}{2a} \)
• \( x = \frac{-b - i \sqrt{\left| \Delta \right|}}{2a} \)

Here, \( i \) is the imaginary unit, defined as \( i^2 = -1 \). Complex solutions mean that the parabola described by the quadratic equation does not intersect the real x-axis.

Standard Form

The standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \). Converting an equation into this form is crucial for solving and analyzing it. In this structure, \( a \) cannot be zero because if it were, the equation would not be quadratic but linear instead.
An easy step-by-step conversion process involves collecting all terms on one side of the equation so that only zero remains on the other. Then arrange the terms by descending powers of \( x \).

• Identify coefficients \( a \), \( b \), and \( c \)
• Check if the equation is properly arranged according to powers of \( x \)
• If needed, rearrange terms to fit \( ax^2 + bx + c = 0 \)

This form is not only practical for solving through the quadratic formula but essential for identifying the discriminant, which clarifies the number and type of solutions.