Problem 17 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$20 y^{2}+17 y-10=0$$

Step-by-Step Solution

Answer

The solutions are \(y = \frac{2}{5}\) and \(y = -\frac{5}{4}\).

Step 1: Examine the quadratic equation

The given equation is \(20y^2 + 17y - 10 = 0\). It's in the standard quadratic form, \(ax^2 + bx + c = 0\), where \(a = 20\), \(b = 17\), and \(c = -10\).

Step 2: Determine the method to solve

Considering that the quadratic involves reasonably complex coefficients, the quadratic formula is a suitable method for solving it: \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Step 3: Calculate the discriminant

The discriminant (\(b^2 - 4ac\)) determines the number and type of solutions. Compute it as follows: \[b^2 - 4ac = 17^2 - 4 \cdot 20 \cdot (-10) = 289 + 800 = 1089\]. Since the discriminant is positive, we'll have two real solutions.

Step 4: Apply the quadratic formula

Substitute \(a = 20\), \(b = 17\), and \(c = -10\) into the quadratic formula: \[y = \frac{-17 \pm \sqrt{1089}}{40}\].

Step 5: Compute the square root and solutions

First, calculate \(\sqrt{1089} = 33\). Now we have:\[y = \frac{-17 \pm 33}{40}\].Calculate the two possible solutions:1. \(y = \frac{-17 + 33}{40} = \frac{16}{40} = \frac{2}{5}\).2. \(y = \frac{-17 - 33}{40} = \frac{-50}{40} = -\frac{5}{4}\).

Step 6: Final solution set

The solutions for the quadratic equation \(20y^2 + 17y - 10 = 0\) are \(y = \frac{2}{5}\) and \(y = -\frac{5}{4}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool to solve quadratic equations of the form \(ax^2 + bx + c = 0\). This means any equation that can be rearranged into this standard format can be tackled using this handy formula: \[y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This formula doesn't come out of thin air; it is derived by completing the square on the general quadratic equation. Once you memorize this magic formula, it can be used to solve any quadratic equation where the coefficients are known.

The formula contains both a 'plus' and 'minus' operation, which implies it can produce two potential solutions. These are called the 'roots' of the equation, as they mark the points where the equation graphically crosses the x-axis.

Discriminant

The discriminant is a crucial component found within the quadratic formula, represented by the expression \(b^2 - 4ac\). It holds the key to understanding the nature of the solutions without actually solving the equation.

• If the discriminant is positive, as it is in this problem with a value of 1089, there are two distinct real solutions.
• If it equals zero, that means there's exactly one real solution, technically known as a repeated root.
• If it's negative, then no real solutions exist, but two complex ones do.

Knowing the discriminant can help you decide how many solutions to expect, which directs the problem-solving pathway you'd choose.

Real Solutions

Real solutions refer to the roots of the quadratic equation that are real numbers. These are the solutions that can be plotted on a number line as opposed to complex or imaginary numbers.

When discussing quadratic equations, real solutions are those that result when the discriminant is zero or positive. For our specific equation, the positive discriminant signified that there were two real solutions.

These solutions often represent meaningful, real-world values, depending on the context of the quadratic equation. For example, in physics, engineering, or economics, these solutions could stand for measurements, times, or quantities that practitioners are trying to compute. In this exercise, we derived the real solutions to be \(y = \frac{2}{5}\) and \(y = -\frac{5}{4}\).

Solving Quadratics

Solving quadratic equations can be approached through several methods, but one of the most foolproof is the quadratic formula due to its consistency and comprehensiveness.

Here's a quick recap of general methods:

• Factoring: This works well when the equation neatly breaks down into simple integer solutions.
• Completing the Square: This is great for deriving the quadratic formula or when the equation is tough to factor.
• Quadratic Formula: This method can solve any quadratic equation, even when it seems impossible to factor or has complex solutions.

In our specific equation, we used the quadratic formula because it offers a systematic way to find solutions, regardless of whether or not the equation can be factored easily.

Spend time practicing with the quadratic formula as part of your toolkit. Familiarity will make problem solving even quicker and more precise.