Problem 24 - Algebra 2 and Trigonometry

Question

When \(b^{2}-4 a c=0,\) is \(a x^{2}+b x+c\) a perfect square trinomial or a constant times a perfect square trinomial? Explain why or why not.

Step-by-Step Solution

Answer

Yes, it is a constant times a perfect square trinomial because the discriminant is zero, resulting in a repeated root.

Step 1: Identify the Discriminant

The discriminant for a quadratic function \( ax^2 + bx + c \) is given by the formula \( b^2 - 4ac \). This value helps to determine the nature of the roots of the quadratic equation. In the given problem, we know \( b^2 - 4ac = 0 \).

Step 2: Roots of the Quadratic Equation

When the discriminant \( b^2 - 4ac = 0 \), it implies that the quadratic equation \( ax^2 + bx + c = 0 \) has exactly one distinct root, meaning it has a repeated root or double root. This can be expressed as \( x = \frac{-b}{2a} \).

Step 3: Recognizing Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that can be rewritten as \( (px + q)^2 \). With a zero discriminant, \( ax^2 + bx + c \) can be expressed as \( a(x + \frac{b}{2a})^2 \), showing that the quadratic is a constant factor \( a \) times a perfect square trinomial.

Step 4: Verification with the Perfect Square Formula

The perfect square trinomial \( (px + q)^2 \) expands to \( p^2x^2 + 2pqx + q^2 \). Here \( p = \sqrt{a} \), \( q = \frac{b}{2\sqrt{a}} \). Simplifying, it relates back to \( ax^2 + bx + c \), confirming it is a constant times a perfect square trinomial.

Key Concepts

Quadratic Function

A quadratic function is a mathematical expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of \( a \). These functions are key in algebra because they describe a wide range of natural phenomena and situations, like projectile motion and area calculations. To better understand them, let's break them down a bit more:

• The term \( ax^2 \) is the quadratic term, responsible for the parabola's curvature.
• The term \( bx \) is the linear term, affecting the slope and direction of the curve.
• The term \( c \) is the constant term, which shifts the graph up or down on the y-axis.

By adjusting these coefficients, you can change the position and shape of the parabola, which allows for great flexibility in modeling situations.

Discriminant

The discriminant of a quadratic function \( ax^2 + bx + c \) is found using the formula \( b^2 - 4ac \). It serves a crucial role in determining the nature of the roots of the quadratic equation. The discriminant tells us how many solutions, or roots, a quadratic equation will have and what type they will be. Here’s how it works:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), the equation has one real root, known as a repeated root.
• If \( b^2 - 4ac < 0 \), the equation has no real roots, but two complex conjugate roots instead.

In the given exercise, we analyze a situation where the discriminant equals zero. This indicates that the quadratic expression forms a perfect square trinomial, leading to a repeated root.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). The process of finding these roots is called solving the quadratic equation. The roots can be determined using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The discriminant \( b^2 - 4ac \) decides the number and type of roots. For example, when \( b^2 - 4ac = 0 \), there's exactly one root. This root is termed a repeated or double root since it's the only solution and appears twice in the context of the quadratic formula:

• The formula simplifies to \( x = \frac{-b}{2a} \), highlighting the root's simplicity.

This repeated root gives the graph a vertex that just touches the x-axis at that point, representing the parabola's minimum or maximum

Repeated Root

A repeated root, also known as a double root, is a root that occurs when a quadratic equation has a discriminant of zero. When this happens, the quadratic function \( ax^2 + bx + c \) does not cross the x-axis twice, but rather just touches it at a single point. The repeated root is calculated as \( x = \frac{-b}{2a} \).Understanding the concept of a repeated root can be quite useful:

• It signifies that the graph of the function is tangent to the x-axis.
• It highlights symmetry, as this point is the vertex of the parabola.
• It implies that the quadratic expression can be rewritten as a square, specifically \( a(x + \frac{b}{2a})^2 \), a key characteristic of perfect square trinomials.

Recognizing repeated roots helps in identifying when a quadratic expression is a constant times a perfect square trinomial, simplifying the process of solving or factoring the equation.