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Chapter 9: Problem 23

Choose the appropriate method to solve the following. $$ 12 x 2-2 x+52=0 $$

Short Answer

Expert verified
The equation has no real solutions due to a negative discriminant.

Step by step solution

01

Identify the Equation Type

The given equation is a quadratic equation, which has the standard form of \( ax^2 + bx + c = 0 \). In this equation, \( a = 12 \), \( b = -2 \), and \( c = 52 \).
02

Calculate the Discriminant

For a quadratic equation \( ax^2 + bx + c = 0 \), the discriminant \( D \) is given by \( D = b^2 - 4ac \). Substitute \( a = 12 \), \( b = -2 \), and \( c = 52 \) into this formula: \( D = (-2)^2 - 4 \times 12 \times 52 \). Calculate these values to get \( D = 4 - 2496 \), which simplifies to \( D = -2492 \).
03

Analyze the Discriminant

Since the discriminant \( D = -2492 \) is less than zero, the quadratic equation does not have real solutions. The solutions would be complex numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant
In the realm of quadratic equations, the discriminant plays a crucial role in determining the nature of the solutions. Every quadratic equation follows the archetype formula: \[ ax^2 + bx + c = 0 \]Here, \( a \), \( b \), and \( c \) are known as the coefficients of the equation. The discriminant, represented by \( D \), can be calculated with the expression:\[ D = b^2 - 4ac \]Understanding the value of the discriminant provides insight into the nature of the roots of the equation:
  • If \( D > 0 \), the equation has two distinct real roots.
  • If \( D = 0 \), the equation has exactly one real root, also known as a repeated or double root.
  • If \( D < 0 \), the equation has no real roots, but instead two complex conjugate roots.
In the provided exercise, the calculated discriminant is \( -2492 \), which is negative, indicating the solutions will be complex numbers.
Complex Numbers
When discussing quadratic equations, complex numbers often emerge as solutions, especially when the discriminant is negative. A complex number takes the form:\[ a + bi \]where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit which is defined by the property \( i^2 = -1 \). When we say a quadratic has complex solutions due to a negative discriminant, it means the roots are not visible on the real number line but exist in the complex plane.Here's how complex solutions manifest:
  • They appear as a pair of conjugates: \( z = a + bi \) and \( \bar{z} = a - bi \).
  • The real part gives the horizontal placement on the complex plane, while the imaginary part represents vertical placement.
Understanding the complex nature of roots aids in solving equations that don't intersect the real line, giving a fuller picture of potential solutions.
Standard Form of Quadratic Equation
The standard form of a quadratic equation is a specific arrangement of terms that allows for straightforward analysis and solution discovery. It is written as:\[ ax^2 + bx + c = 0 \]where:
  • \( a \) is the coefficient of \( x^2 \), indicating the curvature of the parabola created by the equation.
  • \( b \) is the coefficient of \( x \), which influences the parabola's horizontal placement and direction.
  • \( c \) is the constant, dictating the parabola's vertical placement on a graph.
Recognizing a quadratic in standard form is vital as it simplifies solving processes. It allows for the direct application of the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Knowing each element's role within the equation simplifies more complex tasks, such as finding the discriminant or determining solution types. Thus, the standard form is not just a representation but a tool for efficient problem-solving.

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