Problem 29 - Elementary Algebra

Question

Choose the appropriate method to solve the following. $$ y 2-4 y-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( y = 2 + \sqrt{5} \) and \( y = 2 - \sqrt{5} \).

Step 1: Recognize that this is a quadratic equation

The given equation \( y^2 - 4y - 1 = 0 \) is a quadratic equation because it follows the general form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -4 \), and \( c = -1 \).

Step 2: Identify an appropriate method

The quadratic formula method will be used to solve this equation, as it is the most straightforward approach for quadratic equations that do not factor easily.

Step 3: Apply the quadratic formula

The quadratic formula is given by \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute \( a = 1 \), \( b = -4 \), and \( c = -1 \) into this formula.

Step 4: Calculate the discriminant

The discriminant \( b^2 - 4ac \) is calculated as follows: \((-4)^2 - 4(1)(-1) = 16 + 4 = 20\).

Step 5: Solve for y

Substitute the discriminant back into the quadratic formula: \( y = \frac{-(-4) \pm \sqrt{20}}{2 \times 1} \). Simplify to get \( y = \frac{4 \pm \sqrt{20}}{2} \).

Step 6: Simplify the solutions

Calculate the two possible values for \( y \):1. \( y = \frac{4 + \sqrt{20}}{2} = \frac{4 + 2\sqrt{5}}{2} = 2 + \sqrt{5} \)2. \( y = \frac{4 - \sqrt{20}}{2} = \frac{4 - 2\sqrt{5}}{2} = 2 - \sqrt{5} \)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). In these equations:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term.

The formula provides a way to find the values of \( x \) that satisfy the equation. It is expressed as:\[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This equation allows you to calculate two possible solutions for \( y \), due to the \( \pm \) sign, which signifies both addition and subtraction. Make sure to substitute the right values of \( a \), \( b \), and \( c \) from your quadratic equation to find these solutions.

Solving Quadratics

Solving quadratic equations can be approached in different ways, but using the quadratic formula is often the quickest when factoring isn’t easy. Here's a straightforward guide on how to apply it:

• First, identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Use these values in the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• Solve for \( y \) by evaluating the expression both with a plus and a minus in the \( \pm \) part.
• Break it down step by step, calculating the discriminant and then the numerator and denominator separately.
• Finally, simplify the resulting expressions to arrive at the solutions, as demonstrated with \( y = 2 + \sqrt{5} \) and \( y = 2 - \sqrt{5} \).

Discriminant

Understanding the discriminant is crucial in the quadratic formula as it determines the nature of the solutions. The discriminant is the part of the quadratic formula under the square root, \( b^2 - 4ac \). It provides insights into the type of solutions you can expect:

• If the discriminant is positive, as in our example where \( b^2 - 4ac = 20 \), there are two distinct real solutions.
• If it is zero, there is exactly one real solution, which means the parabola touches the x-axis at a single point.
• If negative, there are no real solutions; instead, the solutions are complex numbers, meaning the parabola does not intersect the x-axis at all.

In our specific problem, the discriminant of 20 indicated that two real solutions exist, which we calculated using the quadratic formula.