Problem 18 - Basic Engineering Mathematics

Question

Solve the inequality: \(y^{2}-8 y-10 \geq 0\)

Step-by-Step Solution

Answer

The solution is \(y \in (-\infty, 4 - \sqrt{26}] \cup [4 + \sqrt{26}, +\infty)\).

Step 1: Identify the quadratic inequality

In this exercise, you need to solve the quadratic inequality \(y^2 - 8y - 10 \geq 0\). This means we want to find the values of \(y\) for which this expression is greater than or equal to zero.

Step 2: Solve the equality

First, solve the equation \(y^2 - 8y - 10 = 0\) to find the boundary points, which will be the roots of the quadratic equation. You can do this using the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]where \(a = 1\), \(b = -8\), \(c = -10\).

Step 3: Substitute into the quadratic formula

Substitute \(a = 1\), \(b = -8\), and \(c = -10\) into the quadratic formula:\[ y = \frac{8 \pm \sqrt{(-8)^2 - 4 \times 1 \times (-10)}}{2 \times 1} \]This simplifies to:\[ y = \frac{8 \pm \sqrt{64 + 40}}{2} = \frac{8 \pm \sqrt{104}}{2} \]

Step 4: Simplify the roots

Calculate \(\sqrt{104}\), which simplifies to \(2\sqrt{26}\). So the roots of the equation are:\[ y = \frac{8 \pm 2\sqrt{26}}{2} \]Or further simplified:\[ y = 4 \pm \sqrt{26} \]

Step 5: Determine intervals for testing

The roots \(y = 4 + \sqrt{26}\) and \(y = 4 - \sqrt{26}\) divide the number line into three intervals: 1. \((-\infty, 4 - \sqrt{26})\)2. \((4 - \sqrt{26}, 4 + \sqrt{26})\)3. \((4 + \sqrt{26}, +\infty)\) We need to test these intervals to determine where the inequality holds.

Step 6: Test each interval

Choose a test point from each interval: - For \((-\infty, 4 - \sqrt{26})\), test with \(y = 0\).- For \((4 - \sqrt{26}, 4 + \sqrt{26})\), test with \(y = 4\).- For \((4 + \sqrt{26}, +\infty)\), test with \(y = 10\).Calculate the inequality \(y^2 - 8y - 10\) at each test point to see if the expression is non-negative.

Step 7: Evaluate interval test points

Calculate:- At \(y = 0\), \(y^2 - 8y - 10 = 0^2 - 8(0) - 10 = -10\), not satisfied.- At \(y = 4\), \(y^2 - 8y - 10 = 16 - 32 - 10 = -26\), not satisfied.- At \(y = 10\), \(y^2 - 8y - 10 = 100 - 80 - 10 = 10\), satisfied.Thus, the expression is non-negative in the interval \((4 + \sqrt{26}, +\infty)\).

Step 8: Include boundary points

Check the original inequality's condition at the boundary points. Since the inequality is \(\geq 0\), include the boundary values where the expression equals zero: \(y = 4 - \sqrt{26}\) and \(y = 4 + \sqrt{26}\).

Step 9: Final solution

The solution for \(y^2 - 8y - 10 \geq 0\) is therefore: \[ y \in (-\infty, 4 - \sqrt{26}] \cup [4 + \sqrt{26}, +\infty) \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). By isolating the value of \( x \), you can find the roots of any quadratic equation, which in turn helps with solving quadratic inequalities. The formula is given as follows: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula involves several steps to compute:

• Calculate the discriminant, \( b^2 - 4ac \).
• Take the square root of the discriminant.
• Apply the plus-minus symbol (\( \pm \)) to express both the potential positive and negative roots.
• Divide the entire expression by \( 2a \) to solve for \( x \).

This formula is derived from the process of completing the square and can be used regardless of the nature of the roots (real or complex).

Inequality Solution

Solving a quadratic inequality involves finding where the quadratic expression is greater than or equal to zero, less than, or a combination based on the inequality sign. For example, with the inequality \( y^2 - 8y - 10 \geq 0 \), the goal is to identify all values of \( y \) which make the inequality true. To start solving:

• Rewrite the inequality, if necessary, to have zero on one side.
• Determine the roots of the corresponding quadratic equation \( y^2 - 8y - 10 = 0 \) using the quadratic formula.
• Identify the intervals between the roots where the inequality holds true.
• Through interval testing, check which intervals satisfy the given inequality condition.

Roots of the Equation

The roots of a quadratic equation are the solutions to the equation when set to zero, such as \( y^2 - 8y - 10 = 0 \) in this case. These roots serve as boundary points when testing intervals in inequality solutions. To find these roots, substitute the coefficients \( a = 1 \), \( b = -8 \), and \( c = -10 \) into the quadratic formula:\[y = \frac{8 \pm \sqrt{(-8)^2 - 4 \times 1 \times (-10)}}{2}\] Simplify this to:\[ y = 4 \pm \sqrt{26} \]Here:

• \( y = 4 + \sqrt{26} \)
• \( y = 4 - \sqrt{26} \)

These roots are then used to define the intervals needed for testing the inequality.

Interval Testing

Interval testing is an essential part of solving quadratic inequalities. Once the roots are found, they divide the number line into intervals. In our case with roots \( y = 4 + \sqrt{26} \) and \( y = 4 - \sqrt{26} \), we have three intervals to consider:

• \((-\infty, 4 - \sqrt{26}) \)
• \((4 - \sqrt{26}, 4 + \sqrt{26}) \)
• \((4 + \sqrt{26}, +\infty) \)

For each interval, pick a test point and substitute it into the original inequality \( y^2 - 8y - 10 \geq 0 \) to check if it satisfies the inequality:- Test point in \((-\infty, 4 - \sqrt{26})\) with \(y = 0\) yields a negative result.- Test point in \((4 - \sqrt{26}, 4 + \sqrt{26})\) with \(y = 4\) yields a negative result.- Test point in \((4 + \sqrt{26}, +\infty)\) with \(y = 10\) yields a positive result.This confirms that the inequality holds for the interval \((4 + \sqrt{26}, +\infty)\). Include boundary values if they satisfy the inequality as well.