Problem 50 - Algebra 1

Question

Choose a method to solve the quadratic equation. Explain your choice. $$x^{2}-5 x-1=0$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(x^{2}-5x-1=0\) are \(x1 = \frac{5 + \sqrt{29}}{2}\) and \(x2 = \frac{5 - \sqrt{29}}{2}\).

Step 1: Identify the coefficients

The quadratic equation is of the form \( ax^{2} + bx + c = 0 \). For the given equation \(x^{2}-5x-1=0\), the coefficients are \(a = 1\), \(b = -5\) and \(c = -1\).

Step 2: Insert the coefficients into the quadratic formula

Enter the values into the quadratic formula. This is \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Therefore, we get \(\frac{5 \pm \sqrt{(-5)^2-4*1*(-1)}}{2*1}\).

Step 3: Simplify the equation

Simplify the square root term under the equation \(\sqrt{(-5)^2-4*1*(-1)} = \sqrt{25 + 4} = \sqrt{29}\). This means the equation becomes \(\frac{5 \pm \sqrt{29}}{2}\).

Step 4: Calculate the two solutions

The solutions are then \(x1 = \frac{5 + \sqrt{29}}{2}\) and \(x2 = \frac{5 - \sqrt{29}}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to find solutions for quadratic equations, which are equations in the form of \( ax^2 + bx + c = 0 \). This formula works for any quadratic equation and is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Let's break down its components:

• \( -b \): Represents the negation of the coefficient of \( x \), which helps calculate the turning point.
• \( \pm \sqrt{b^2 - 4ac} \): Indicates that there may be two possible solutions (one by addition and one by subtraction) which are derived from the nature of quadratic equations.
• \( 2a \): Is the denominator that normalizes the whole equation according to the coefficient of \( x^2 \).

The formula is derived from completing the square method and is a universally accepted standard for solving all types of quadratic equations. This makes it incredibly versatile and widely used in mathematics.

Coefficients

Coefficients are the numbers or parameters of a term that appear in front of variables in any polynomial expression. In the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are \( a \), \( b \), and \( c \). These coefficients determine the shape and position of the parabola represented by the quadratic equation.

• \( a \): Known as the leading coefficient, it affects the curvature of the parabola. A larger \( a \) makes the parabola steeper.
• \( b \): Influences the position of the axis of symmetry of the parabola and impacts its horizontal orientation.
• \( c \): This is the constant term which tells us where the parabola crosses the y-axis.

Determining these coefficients is the starting point when using the quadratic formula to solve an equation. They are vital as they directly influence the values under the square root in the quadratic formula.

Solution Method

When facing the task of solving a quadratic equation, choosing the right method is crucial for ease and accuracy. Several solution methods exist, including:

• Quadratic Formula: As demonstrated previously, this method is very reliable and applies to all quadratic equations.
• Factoring: Suitable for equations where factors can be easily guessed or found, making it quicker for simple equations.
• Completing the Square: Useful in deriving the quadratic formula or when transformations of the equation make it simple to solve.
• Graphing: Provides a visual representation and is useful for estimating solutions.

For the given equation \(x^2 - 5x - 1 = 0\), the quadratic formula was chosen due to its straightforward approach and ability to handle the lack of obvious factors.

Simplifying Radicals

Simplifying radicals is a critical step, especially when using the quadratic formula, as it directly influences the accuracy of the results. Radicals appear when dealing with the square root component of the quadratic formula. To simplify radicals:

• Calculate the expression inside the square root, known as the discriminant, \( b^2 - 4ac \).
• Assess if the resulting number can be reduced into a simpler radical form. For example, \( \sqrt{25} \) simplifies to \( 5 \).
• If not a perfect square, as in \( \sqrt{29} \), it remains a radical and needs the proper mathematical notation of "plus/minus" in the quadratic formula \( \pm \).

In our example, \( \sqrt{29} \) cannot be simplified further, so it is kept as a part of the final solution expression. Proper management and simplification of radicals ensure correct solutions are reached for any quadratic equation.