Problem 65 - 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 the Coefficients

The general form of a quadratic equation is \( ax^2 + bx + c = 0 \). In the given equation, \( x^2 - 6x + 1 = 0 \), we can identify the coefficients as follows: \( a = 1 \), \( b = -6 \), and \( c = 1 \).

Step 2: Recall the Discriminant Formula

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the formula \( \Delta = b^2 - 4ac \). This formula helps in determining the nature of the roots of the quadratic equation.

Step 3: Calculate the Discriminant

Substitute the identified values into the discriminant formula: \( \Delta = (-6)^2 - 4(1)(1) \). Simplify to find \( \Delta = 36 - 4 = 32 \).

Step 4: Interpret the Discriminant

The value of the discriminant \( \Delta = 32 \) is greater than zero. This indicates that the quadratic equation has two distinct real solutions.

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a mathematical expression of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents an unknown variable. Quadratic equations are fundamental in algebra and appear frequently in various fields such as physics, engineering, and economics.
A key feature of quadratic equations is their characteristic U-shaped graph, called a parabola. The graph can open upwards or downwards depending on the sign of the coefficient \( a \).The Main Components of a Quadratic Equation:

• Coefficient \( a \): Determines the direction of the parabola. If \( a > 0 \), it opens upwards. If \( a < 0 \), it opens downwards.
• Coefficient \( b \): Influences the position of the vertex of the parabola along the x-axis.
• Coefficient \( c \): Represents the y-intercept, the point where the parabola crosses the y-axis.

The solutions of a quadratic equation, known as roots, can be found by various methods including factoring, completing the square, or using the quadratic formula.

Exploring Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation within the set of real numbers. Depending on the quadratic equation, the number of real solutions can vary.
Solving these equations is crucial, especially when working on problems that require an understanding of where a function intersects the x-axis.There are three possibilities concerning real solutions:

• Two distinct real solutions: This occurs when the parabola intersects the x-axis at two points.
• One real solution: Occurs when the vertex of the parabola touches the x-axis.
• No real solutions: The parabola does not touch the x-axis at all.

The discriminant plays a crucial role in determining which of these cases holds for a given quadratic equation.

Nature of Roots and the Discriminant

The nature of the roots of a quadratic equation refers to the type and quantity of its solutions. The discriminant, represented by \( \Delta = b^2 - 4ac \), is a component of the quadratic formula that provides insight into this aspect.
By calculating the discriminant, we can predict the roots without solving the equation completely.The discriminant tells us:

• \( \Delta > 0 \): Two distinct real roots. The parabola intersects the x-axis in two different places.
• \( \Delta = 0 \): One real root, indicating that the vertex of the parabola is on the x-axis.
• \( \Delta < 0 \): No real roots since the parabola does not touch the x-axis.

For the equation \( x^2 - 6x + 1 = 0 \), we calculated a discriminant of 32, meaning it has two distinct real solutions.

Using the Quadratic Formula

The quadratic formula offers a reliable method for finding the roots of any quadratic equation. This formula can always be applied as long as you know the coefficients \( a \), \( b \), and \( c \).
The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To use the quadratic formula, you substitute the coefficients into the formula and simplify. The "\( \pm \)" symbol indicates that there are typically two solutions — provided that the discriminant is non-negative.Steps to Use the Quadratic Formula:

• Identify \( a \), \( b \), and \( c \) from the quadratic equation.
• Calculate the discriminant \( \Delta = b^2 - 4ac \).
• Substitute \( a \), \( b \), \( c \), and \( \Delta \) into the formula.
• Simplify to find the values of \( x \).

Remember, if the discriminant is greater than zero, you'll obtain two distinct solutions, just as we found for \( x^2 - 6x + 1 = 0 \). If the discriminant is less than zero, there are no real solutions using the quadratic formula.