Question

LOGICAL REASONING Consider the equation \(a x^{2}+b x+c=0\) and use the quadratic formula to justify the statement. If \(b^{2}-4 a c\) is negative, then the equation has no real solution.

Short answer

The value of the discriminant \(b^2 - 4ac\) determines the nature of the solutions of a quadratic equation. If it's negative, the square root part of the quadratic formula -\(\sqrt{b^2-4ac}\)- will be the square root of a negative number, which is not a real quantity. Therefore, there are no real solutions, justifying the statement.

Step-by-step solution

Understand the quadratic formula

The quadratic formula \(x = \frac{-b ± \sqrt{b^2-4ac}}{2a}\) is the solution for the quadratic equation \(ax^2 + bx + c = 0\), where \(x\) is the value of the unknown variable that solves the equation. The term under the square root (b^2 - 4ac) is known as the discriminant.

Understand the discriminant

The discriminant can be used to determine the number and type of solutions. If the discriminant is greater than zero, then there are two distinct real solutions. If it's equal to zero, then there's exactly one real solution. If it's less than zero (negative), then there are no real solutions.

Apply understanding to the given statement

According to the behavior of the discriminant, if \(b^2 - 4ac\) is negative, the square root of a negative number is not a real quantity (since no real number multiplied by itself can give a negative result). Hence in this case, using the quadratic formula won't yield any real solutions for \(x\), justifying the statement.

Key concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree, meaning it involves the variable \(x\) raised to the power of two.

The most general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The solutions to this equation are the values of \(x\) that make the equation true. These solutions are also known as the 'roots' of the equation.

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), provides a straightforward method for finding these roots. This formula is derived from the process of completing the square and can solve any quadratic equation, even if the roots are complex numbers. It is a powerful tool that allows students to understand the behavior and characteristics of quadratic functions.

Discriminant Analysis

The discriminant in the context of quadratic equations is the part of the quadratic formula under the square root, expressed as \(b^2 - 4ac\). This value is fundamental in discriminant analysis as it reveals the nature of the solutions for the quadratic equation.

If the discriminant is positive, there are two distinct real roots. If it is zero, the equation has exactly one real root, indicating that both roots are the same. Conversely, if the discriminant is negative, no real number solutions exist because the square root of a negative number is not a real number.

Understanding the discriminant is crucial for predicting the types of solutions without actually calculating them. It provides a quick way to assess whether the solutions are real and distinct, real and identical, or complex.

Real Solutions of Quadratic Equations

For a quadratic equation to have real solutions, the discriminant must be zero or positive. Real solutions are the x-values for which the quadratic expression yields zero, corresponding to the points where the graph of the quadratic function crosses the x-axis.

When the discriminant is positive, \(b^2 - 4ac > 0\), the quadratic equation has two distinct real solutions. These are the points at which the parabola intersects the x-axis at two points. If the discriminant is zero, the parabola touches the x-axis at exactly one point, known as a 'double root' or a 'repeated root'.

This concept emphasizes the connection between the algebraic solutions of quadratic equations and their geometric representation on a graph. Students can see how changing the values of \(a\), \(b\), and \(c\) affects both the location and number of real solutions.

Complex Numbers

When the discriminant of a quadratic equation is negative, the equation has no real solutions. Instead, it has complex solutions. Complex numbers are numbers that comprise a real part and an imaginary part, and are typically written in the form \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the square root of -1.

Negative discriminants lead to the square root of a negative number, introducing the imaginary unit \(i\) into the solutions. Such cases produce a pair of complex conjugate roots, which means they have the same real part but opposite imaginary parts.

The concept of complex numbers broadens the range of possible solutions for equations beyond the real numbers, a significant step in math that has applications in various fields, including engineering, physics, and computer science. Students who grasp this fundamental concept can better understand the full spectrum of algebraic solutions.