Problem 163 - College Algebra

Question

If a quadratic equation has imaginary solutions, how is this shown on the graph of \(y=a x^{2}+b x+c ?\)

Step-by-Step Solution

Answer

The imaginary solutions of a quadratic equation are represented on its graph as the absence of any intersection with the x-axis. The whole graph stays above (if a > 0) or below (if a < 0) the x-axis.

Step 1: Understanding Quadratic Equations

A quadratic equation in the form of \(y=ax^{2}+bx+c\) where \(a\), \(b\), and \(c\) are constants and \(a ≠ 0\). The roots of this equation can be found using the quadratic formula \((-b±\sqrt{b^2-4ac})/2a\). The term inside the square root is called the discriminant.

Step 2: Understanding Imaginary Solutions

When we discuss imaginary solutions of an equation, it refers to the outcomes where the part under the square root (the discriminant) in the quadratic formula becomes negative. This happens when \(b^2 - 4ac < 0\). That's because the square root of a negative number leads to imaginary solutions. Consequently, it gives us two complex roots which are conjugate of each other.

Step 3: Relating Imaginary Solutions to the Quadratic Graph

From a graphical standpoint, if a quadratic equation has real roots, it will intersect the x-axis at points representing these roots. In case of imaginary roots, these intersection points with x-axis don't exist because an imaginary number is not on the real number line, which is represented by the x-axis. Therefore, the graph of the function \(y=ax^{2}+bx+c\) will not intersect with the x-axis if the equation's solutions are imaginary. Instead, the graph of the function either lies entirely above (if a > 0) or below (if a < 0) the x-axis.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally expressed in the standard form as y = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. These equations are pivotal in algebra and appear frequently across various fields of math and science.

A quintessential property of quadratic equations is that they can have up to two real solutions, known as the roots of the equation. These solutions are where the graph of the equation crosses the x-axis, which can be visualized when graphing the function. Calculating these roots is made possible by employing methods such as factoring, completing the square, or more universally, the quadratic formula.

Discriminant

The discriminant is a key feature within the quadratic formula and is represented by the symbol Δ in the expression Δ = b^2 - 4ac. It informs us about the nature of the roots without actually solving the equation.

The value of the discriminant can result in three different scenarios:

• If Δ > 0, the quadratic equation has two distinct real roots.
• If Δ = 0, the equation has exactly one real root, also known as a repeated root.
• If Δ < 0, which brings us to imaginary solutions, as there are no real roots.

Complex Roots

When a quadratic equation's discriminant is negative (Δ < 0), it indicates the presence of complex roots. These roots take the form of a ± bi, where a and b are real numbers, and i represents the imaginary unit – the square root of -1.

Complex roots occur in conjugate pairs, which means if a + bi is a root, then a - bi is also a root. This concept is deeply rooted in the Fundamental Theorem of Algebra, which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This is pertinent in the study of quadratic equations as it guarantees two roots (either real or complex) for every quadratic polynomial.

Graphing Quadratic Functions

The graph of a quadratic function is a U-shaped curve called a parabola. When graphing the quadratic function y = ax^2 + bx + c, the orientation (upward or downward) of the parabola is determined by the sign of a. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.

Moreover, as the value of the discriminant reveals the nature of the roots, the graph provides a visual representation of this. A positive discriminant corresponds to the parabola crossing the x-axis twice, while a discriminant of zero means that the vertex of the parabola touches the x-axis. Importantly, an imaginary solution, indicated by a negative discriminant, results in the parabola not intersecting the x-axis at all. It either lies entirely above the x-axis (if a > 0) or below it (if a < 0), underscoring the connection between the discriminant, the roots, and the graphical representation of quadratic functions.