Problem 56 - Algebra 1: Concepts and Skills

Question

Decide whether the statement is true or false. If it is true, give a reason. If it is false, give a counterexample. \(x^{2}=c\) has no real solution when \(c<0\)

Step-by-Step Solution

Answer

The statement is true. The equation \(x^{2}=c\) has no real solutions when \(c<0\), because by analyzing the discriminant, when \(c\) is less than zero, the solution to \(x^{2}=c\) will be a complex number, not a real one.

Step 1: Understanding the Statement

The given problem statement indicates that the equation \(x^{2}=c\) has no real solution when \(c<0\). In other words, for any negative number \(c\), the equation \(x^{2}=c\) would not have real solutions.

Step 2: Analyzing the Conditions

A quadratic equation has no real solutions under one key condition: when the discriminant of the equation is less than zero. Discriminant is calculated with the formula \(b^{2}-4ac\) for a quadratic equation of the form \(ax^{2} + bx + c = 0\). If the discriminant is less than zero, it means there are no real roots or solutions for the equation.

Step 3: Applying to the Given Statement

For our equation, \(x^{2}=c\), we can rewrite it as \(x^{2}-c=0\), which is in the form of \(ax^{2} + bx + c = 0\) where \(a=1\), \(b=0\), and \(c=-c\). The discriminant of our equation becomes \(b^{2}-4ac = 0^{2}-4(1)(-c) = 4c\). Thus, when \(c<0\), the discriminant is less than zero. Hence, there are no real solutions to the equation \(x^{2}=c\) where \(c\) is a negative number.

Key Concepts

Quadratic Discriminant

The quadratic discriminant is a critical concept when analyzing the nature of the roots of quadratic equations. When we have a quadratic equation in the standard form, which is \( ax^{2} + bx + c = 0 \), we can determine the type of solutions the equation will have by calculating its discriminant using the formula \( b^{2}-4ac \).

The discriminant provides us with valuable information:

• \(\text{If } b^{2}-4ac > 0\), the equation has two distinct real solutions.
• \(\text{If } b^{2}-4ac = 0\), the equation has exactly one real solution, also known as a repeated or double root.
• \(\text{If } b^{2}-4ac < 0\), the equation has no real solutions, and instead, it has two complex solutions.

Understanding the discriminant helps students predict the result without actually solving the equation. In the case of our problem, with \(c < 0\), we see that the discriminant is negative, hence no real solutions exist.

Complex Numbers

When a quadratic equation does not have real solutions, it is often because the roots involve complex numbers. Complex numbers are numbers that have a real part and an imaginary part, and are typically written in the form \( a + bi \), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \( i^{2} = -1 \).

In the context of quadratic equations, when the discriminant is negative, the solutions will involve square roots of negative numbers, which is not possible in the real number system. For example, solving \( x^{2} = -1 \) leads to \( x = \pm\sqrt{-1} \) or \( x = \pm i \). These are complex solutions. Grasping the concept of complex numbers is essential for understanding the full spectrum of solutions that quadratic equations can have.

Solving Quadratic Equations

The process of solving quadratic equations can be done using several methods, including factoring, completing the square, using the quadratic formula, and graphing. The quadratic formula, \( x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a} \), is derived from the process of completing the square and allows one to find the solutions by plugging the coefficients \(a\), \(b\), and \(c\) into the formula.

When using the quadratic formula, if the discriminant is negative, it indicates the presence of complex solutions, while a non-negative discriminant corresponds to real solutions. For students, the quadratic formula is a powerful tool that consolidates their understanding of the discriminant and the overall process of finding roots, whether they're real or complex.

Nature of Roots

The nature of the roots of a quadratic equation—whether they are real or complex, distinct or repeated—significantly impacts the graph and the solution set of the equation. Real roots imply that the graph of the quadratic equation will intersect the x-axis, while complex roots suggest that the graph does not intersect the x-axis at all.

In cases where we have two real and distinct roots, the graph will touch the x-axis at two points. If there's a single real root, it means the vertex of the parabola is on the x-axis, and for complex roots, the parabola lies entirely above or below the x-axis (depending on the coefficient \(a\)). This distinction is crucial for visualizing equations and for understanding how the discriminant influences the graph of a quadratic equation.