Problem 61 - Algebra and Trigonometry

Question

Use the discriminant to determine the number of real solutions of the equation. Do not solve the equation. \(x^{2}-6 x+1=0\)

Step-by-Step Solution

Answer

The equation has two distinct real solutions.

Step 1: Identify Coefficients

First, identify the coefficients a, b, and c from the quadratic equation of the form \(ax^2 + bx + c = 0\). For the equation \(x^2 - 6x + 1 = 0\), the coefficients are: \(a = 1\), \(b = -6\), \(c = 1\).

Step 2: Write Discriminant Formula

Recall the discriminant formula for a quadratic equation, which is given by \(b^2 - 4ac\). This formula helps determine the nature of the roots of the equation.

Step 3: Calculate the Discriminant

Substitute the values of a, b, and c into the discriminant formula: \((-6)^2 - 4(1)(1)\). Calculate the discriminant: \(36 - 4 = 32\).

Step 4: Analyze the Discriminant

A positive discriminant indicates two distinct real solutions, zero indicates one real solution (a repeated root), and a negative discriminant indicates no real solutions. Since \(32 > 0\), there are two distinct real solutions.

Key Concepts

Quadratic Equation

A quadratic equation is an essential concept in algebra and it takes the form \(ax^2 + bx + c = 0\). This formula represents a parabola on a graph. Here are the main parts of a quadratic equation:

• The term \(ax^2\) is called the quadratic term, where \(a\) is the coefficient.
• The term \(bx\) is the linear term, with \(b\) as its coefficient.
• Lastly, \(c\) is the constant term without any variables.

To effectively use the quadratic equation, one should always identify the coefficients \(a\), \(b\), and \(c\) properly. They play a crucial role when applying methods like factoring, the quadratic formula, or completing the square to find solutions.

Real Solutions

When trying to find the number of real solutions of a quadratic equation, the discriminant is vital. The discriminant is part of the quadratic formula and is expressed as \(b^2 - 4ac\).

• If the discriminant is greater than zero, it results in two distinct real solutions.
• If the discriminant equals zero, it gives one real solution, which is referred to as a repeated root because the parabola touches the x-axis at exactly one point.
• If the discriminant is less than zero, there are no real solutions, meaning the parabola does not intersect the x-axis.

Understanding the discriminant helps predict the graph's intersections with the x-axis without actually solving the equation.

Roots of Equation

The roots of a quadratic equation refer to the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). These roots, or solutions, can be:

• Real: When the graph intersects the x-axis at these points.
• Complex: When the discriminant is negative, leading to imaginary solutions as they involve square roots of negative numbers.

In the given exercise, the discriminant was calculated to be 32, meaning there are two distinct real roots. The roots of the equation directly correlate with the nature of its solutions, making it a crucial part in solving quadratics effectively.