Problem 46 - Algebra 2

Question

OPEN ENDED Give an example of a quadratic equation with a double root, and state the relationship between the double root and the graph of the related function.

Step-by-Step Solution

Answer

The quadratic equation \(x^2 - 4x + 4 = 0\) has a double root at \(x = 2\), making the graph a parabola that just touches the x-axis at one point.

Step 1: Define a Double Root

A double root occurs when a quadratic equation has a repeated root, meaning the discriminant equals zero, \((b^2 - 4ac = 0)\). This leads to only one solution for the quadratic equation \(ax^2 + bx + c = 0\).

Step 2: Create an Example Equation

Consider the quadratic equation \(x^2 - 4x + 4 = 0\). Expanding, this simplifies to \((x - 2)^2 = 0\), indicating a double root at \(x = 2\).

Step 3: Verify the Double Root

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), substitute \(a = 1\), \(b = -4\), and \(c = 4\). The root is \(x = \frac{4 \pm 0}{2} = 2\). Thus, the double root is confirmed as \(x = 2\).

Step 4: Describe the Graph

The graph of the quadratic function \(y = x^2 - 4x + 4\) is a parabola that touches the x-axis at the double root \(x = 2\), representing a point of tangency. This means the parabola does not cross the x-axis and only "kisses" it at \(x = 2\).

Key Concepts

Double Root

A double root in a quadratic equation is a situation where the equation has the same solution twice. This intriguing concept occurs when the discriminant, a special part of the equation given by the formula \(b^2 - 4ac\), equals zero. When this happens, the equation \(ax^2 + bx + c = 0\) has only one unique solution, which it must solve "twice." This is why it's often referred to as having a repeated or double root. A simple example is the quadratic equation \(x^2 - 4x + 4 = 0\). When expanded, it reveals its double root structure as \((x - 2)^2 = 0\), indicating that \(x = 2\) is the root that repeats. This concept helps in understanding the nature of the equations related to their graphical representation.

Discriminant

The discriminant of a quadratic equation is an essential part of the quadratic formula that helps determine the nature of the equation’s roots. It is denoted as \(b^2 - 4ac\). A key point to remember is:

• If the discriminant is positive, the quadratic has two distinct real roots.
• If the discriminant is zero, there is exactly one real root, also known as a double root or repeated root.
• If the discriminant is negative, the quadratic has no real roots, only complex ones.

When you have a zero discriminant, it indicates a perfectly balanced square in the related function, signifying the double root condition. Hence, it acts as a quick-check tool to predict root behavior without solving the equation completely.

Quadratic Formula

The quadratic formula is a universal solution technique used for finding the roots of a quadratic equation. The formula itself is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It provides solutions for the equation \(ax^2 + bx + c = 0\). The miracle of this formula lies in its ability to solve any quadratic equation, offering two potential solutions due to the "\(\pm\)" sign, which accounts for both positive and negative solutions. When the discriminant part, \(b^2 - 4ac\), equals zero, the formula simplifies to \(x = \frac{-b}{2a}\), revealing the double root scenario where both potential solutions collapse into one single repeated solution.

Parabola

A parabola is the graphical representation of a quadratic function, usually expressed as \(y = ax^2 + bx + c\). This curve is symmetrical and can open either upwards or downwards, depending on the sign of the coefficient \(a\). The vertex of the parabola is its peak or lowest point, and its position is closely related to the roots of the quadratic function. In the case of a double root, the shape of the parabola is such that the vertex is exactly on the x-axis. This unique tangential touch to the x-axis implies that the graph doesn't cross it but merely "kisses" it at just one point. This point is both the vertex and the x-intercept (or root) of the function. Understanding this helps make sense of the symmetry and the balance present in quadratic graphs.

X-axis Intersection

The x-axis intersection of a quadratic graph is where the parabola meets or touches the x-axis. In standard quadratic problems, finding these intersections is equivalent to solving the quadratic equation for its roots. However, for a quadratic with a double root, the x-axis intersection occurs uniquely at one point. In this scenario, the parabola doesn’t cross the x-axis but merely touches it at one precise point, the double root. This is evident in the equation \(x^2 - 4x + 4 = 0\) with the graph touching the x-axis at \(x = 2\). Parabolas with such intersections visually underscore the special condition of having a repeated root and help to concretize the relationship between algebraic solutions and geometric interpretations.