Problem 38 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$8 n^{2}-6 n-5=0$$

Step-by-Step Solution

Answer

The solutions are \(n = \frac{5}{4}\) and \(n = -\frac{1}{2}\).

Step 1: Identify the quadratic equation format

The given equation is in the standard quadratic form \(ax^2 + bx + c = 0\) where \(a = 8\), \(b = -6\), and \(c = -5\).

Step 2: Use the quadratic formula

Since factoring directly may be challenging, use the quadratic formula, which is \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this case, \(a = 8\), \(b = -6\), and \(c = -5\).

Step 3: Calculate the discriminant

The discriminant is the part under the square root in the quadratic formula: \(b^2 - 4ac\). Substitute the values to get \((-6)^2 - 4(8)(-5) = 36 + 160 = 196\).

Step 4: Solve using the quadratic formula

Substitute the determined values into the quadratic formula: \[n = \frac{-(-6) \pm \sqrt{196}}{2 \times 8}\]This simplifies to: \[n = \frac{6 \pm 14}{16}\]

Step 5: Compute the roots

Calculate both possible values for \(n\): 1. \(n = \frac{6 + 14}{16} = \frac{20}{16} = \frac{5}{4}\) 2. \(n = \frac{6 - 14}{16} = \frac{-8}{16} = -\frac{1}{2}\)

Step 6: Confirm the solution

Verify the roots by substituting back into the original equation to ensure both are valid. Both roots satisfy the original equation.

Key Concepts

Quadratic Formula

The quadratic formula is a key mathematical tool used to find the solutions to quadratic equations - equations of the form \(ax^2 + bx + c = 0\). Here:

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

The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the possible values of \(x\) where the quadratic equation equals zero. The symbol \(\pm\) indicates that there are generally two solutions: one for addition and one for subtraction. Applying this formula is quite handy, especially when other methods, such as factoring, prove difficult.
It is important to ensure correct substitution of coefficients \(a\), \(b\), and \(c\) into the formula for accurate calculations.

Discriminant

The discriminant is a critical component of the quadratic formula situated under the square root symbol: \(b^2 - 4ac\). The value of the discriminant provides insightful information:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one unique real root (the roots are identical).
• If the discriminant is negative, the roots are complex or imaginary numbers, meaning they do not intersect the real number line.

In the context of the given example, the discriminant is calculated as \(196\), a positive number. Therefore, we anticipate two different real solutions when solving the equation. This part of the quadratic formula is essential since it reveals the nature of the roots before even solving them.

Factoring Techniques

Factoring is another method to solve quadratic equations, often preferred because it can be simpler and more intuitive than the quadratic formula where applicable. The idea is to express the quadratic equation as a product of linear factors.
However, factoring relies on recognizing the equation in a suitable form, which is not always straightforward. For example, certain quadratic equations cannot be easily factored without advanced techniques or trial and error. In such situations, the quadratic formula is a more reliable choice.
To factor our example equation, you would typically look for numbers that multiply to \(ac\) (\(8 \times -5 = -40\)) and add to \(b\) (\(-6\)). If such numbers can't be found easily, as in this case, applying the quadratic formula is often a faster and surer path to the solution.