Problem 49 - Elementary Algebra

Question

Use the discriminant to determine the number and type of solutions. $$ 4 x 2-4 x+1=0 $$

Step-by-Step Solution

Answer

The equation has one real repeated root (two identical solutions).

Step 1: Identify the coefficients

In the quadratic equation in the form \(ax^2 + bx + c = 0\), identify the coefficients. Here, \(a = 4\), \(b = -4\), and \(c = 1\).

Step 2: Calculate the discriminant

Use the formula for the discriminant, \(D = b^2 - 4ac\). Substitute the identified coefficients into the formula: \(D = (-4)^2 - 4\times4\times1 = 16 - 16 = 0\).

Step 3: Interpret the discriminant

Since the discriminant \(D = 0\), there is exactly one real solution, which is a repeated root.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, usually in the form \( ax^2 + bx + c = 0 \). It's characterized by the presence of an \(x^2\) term, meaning the highest exponent of the variable \(x\) is 2. This gives the equation its name, as "quad" means square. In the given exercise, the quadratic equation \( 4x^2 - 4x + 1 = 0 \) follows this standard form.

• \(a\), \(b\), and \(c\) are coefficients. They represent known numbers, which you determine from the specific equation given.
• In our exercise, \(a = 4\), \(b = -4\), and \(c = 1\).

Understanding the standard form of a quadratic equation is important because it allows us to perform further analyses, like finding the number and types of solutions using the discriminant.

Number of Solutions

The number of solutions of a quadratic equation is determined by the discriminant, \( D \), which is calculated using the expression \( D = b^2 - 4ac \). The discriminant tells us how many times the parabola intersects the x-axis, which correlates to the roots or solutions of the equation.

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one repeated real solution, meaning the parabola touches the x-axis at exactly one point.
• If \( D < 0 \), the equation has no real solutions, but rather two complex solutions.

In this exercise, the discriminant is calculated to be 0, which means there is one solution.

Types of Solutions

The types of solutions for a quadratic equation are determined by the value of the discriminant. This understanding helps predict if solutions are real or complex, and if real, whether they are identical or distinct.

• Distinct real solutions: When \( D > 0 \), the roots are real and different. This means the graph of the quadratic equation crosses the x-axis at two points.
• Repeated real solution: When \( D = 0 \), the equation has one real solution repeated twice. This is also known as a root of multiplicity two; the graph of the equation just touches the x-axis.
• Complex solutions: When \( D < 0 \), there are no real solutions. Instead, the solutions are complex, indicating the parabola does not intersect the x-axis at all.

In the given problem, since \( D = 0 \), the solution to the equation is a repeated real root, implying the quadratic equation has exactly one solution that occurs twice at the same value of \(x\). This kind of solution is sometimes called a double root.