Problem 80 - Algebra and Trigonometry

Question

Compute the discriminant. Then determine the number and type of solutions for the given equation. $$ 3 x^{2}=2 x-1 $$

Step-by-Step Solution

Answer

The discriminant is -8. Therefore, the given equation has two complex roots.

Step 1: Rewrite Equation

Rewrite the given equation to the form \( ax^{2} + bx + c = 0 \). So, \( 3x^{2}-2x +1 = 0 \). In this equation, a = 3, b = -2 and c = 1.

Step 2: Compute the Discriminant

The discriminant is given by the formula \( D = b^{2} - 4ac \) where a=3, b=-2, c=1. Plugging in these values, we get: \( D = (-2)^{2} - 4*3*1 \) = 4 - 12 = -8

Step 3: Determine the number and type of solutions

If D>0, then there are 2 real roots. If D=0, there are 2 equal real roots. If D<0, there are 2 complex roots. As the value of D in the equation calculated is -8, which is less than 0, we determine that there are 2 complex roots.

Key Concepts

Quadratic Equations

Quadratic equations are fundamental elements of algebra that appear in various mathematical contexts. They are second-order polynomial equations in one variable and have the general form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) represent constants, and \( a \) is not equal to zero. The solutions to these equations, known as roots, can be real or complex and are crucial for graphing parabolas and understanding the behavior of quadratic functions.

Quadratic equations can model many real-world scenarios such as projectile motion, the area of rectangles, and market equilibrium in economics. They are characterized by their distinctive 'U-shaped' graphs that can either open upwards or downwards, depending on whether \( a \) is positive or negative. The vertex of the parabola, the maximum or minimum point of the graph, plays a pivotal role in understanding these equations too.

Types of Roots

The types of roots you can expect from a quadratic equation depend on the discriminant, which is part of the quadratic formula used to solve these equations. The discriminant, symbolized as \( D \), is calculated from the coefficients of the quadratic equation using the formula \( D = b^2 - 4ac \).

The possible types of roots include:

• Real and Distinct: If \( D > 0 \), the equation has two distinct real roots.
• Real and Equal: If \( D = 0 \), the equation has one real root, but it is counted twice—also known as a repeated or double root.
• Complex: If \( D < 0 \), the equation does not have real roots. Instead, it has a pair of complex roots, which are conjugates of each other.

The nature of the roots directly affects the graph of the quadratic equation and can provide insight into the real-world phenomena modeled by the equation.

Complex Roots

Complex roots arise when the discriminant of a quadratic equation is negative. This occurs because the square root of a negative number is not defined within the set of real numbers, introducing the need for complex numbers. Complex numbers have the form \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the square root of \( -1 \).

When you encounter a negative discriminant, it's a sign that your quadratic equation has a pair of complex roots. These roots are always in conjugate pairs, meaning if one root is \( a + bi \), the other is \( a - bi \). Despite sounding abstract, complex numbers are used in many fields, including engineering and physics, to solve problems involving oscillations and waves and in electrical circuits to represent impedance and phase differences.

Solving Quadratic Equations

Solving quadratic equations can be approached through several techniques. The most appropriate method often depends on the form of the equation and the context in which you're working. Some methods are:

• Factoring: Expressing the quadratic equation as the product of two binomials, when possible, and using the zero product property to find the roots.
• Completing the Square: Transforming the equation so that one side is a perfect square trinomial.
• Quadratic Formula: Using the formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) where \( D \) is the discriminant. This formula can determine all types of roots, real or complex, and is a reliable method when other techniques are challenging to apply.
• Graphing: Visualizing the function on a coordinate plane to find the x-intercepts, if they exist, which correspond to the roots of the equation.

The choice of method may affect the ease of finding the solutions, but it's always important to check your answers by substituting the roots back into the original equation to verify they satisfy it.