Problem 36 - College Algebra with Modeling and Visualization

Question

Solve the inequality. $$ -x^{2}-2 x+8>0 $$

Step-by-Step Solution

Answer

The solution is the interval \((-4, 2)\).

Step 1: Find the Roots

To solve the inequality \[-x^2 - 2x + 8 > 0\],first, find the roots of the equation \[-x^2 - 2x + 8 = 0\]by rewriting it as:\[x^2 + 2x - 8 = 0\].Use the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},\]where \(a = 1\), \(b = 2\), \(c = -8\). Substitute the values:\[b^2 - 4ac = 2^2 - 4(1)(-8) = 4 + 32 = 36,\]\[x = \frac{-2 \pm \sqrt{36}}{2} = \frac{-2 \pm 6}{2}.\]Thus, \(x = 2\) and \(x = -4\).

Step 2: Analyze the Sign Intervals

With roots \(x = 2\) and \(x = -4\), the real line is divided into three intervals: \((-\infty, -4)\), \((-4, 2)\), and \((2, \infty)\). To determine where the inequality \(-x^2 - 2x + 8 > 0\)holds, choose a test point from each interval and substitute it into the inequality. For \((-\infty, -4)\), substitute \(x = -5\): \[-((-5)^2) - 2(-5) + 8 = -25 + 10 + 8 = -7 < 0.\]For \((-4, 2)\), substitute \(x = 0\): \[-(0)^2 - 2(0) + 8 = 8 > 0.\]For \((2, \infty)\), substitute \(x = 3\): \[-(3)^2 - 2(3) + 8 = -9 - 6 + 8 = -7 < 0.\]The inequality holds in the interval \((-4, 2)\).

Step 3: Conclusion

The solution to the inequality \(-x^2 - 2x + 8 > 0\)is the interval \((-4, 2)\),where the quadratic expression is greater than zero.

Key Concepts

Quadratic Formula

The quadratic formula is a valuable tool for solving quadratic equations, commonly expressed as \( ax^2 + bx + c = 0 \). By plugging the coefficients \( a \), \( b \), and \( c \) into the formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \),

we can easily find the solutions, or roots, of the equation. This methodology offers a systematic approach, accommodating both real and complex roots.
In our problem, once we rewrite the inequality \(-x^2 - 2x + 8 = 0\) as \(x^2 + 2x - 8 = 0\), we use \( a = 1 \), \( b = 2 \), and \( c = -8 \). By inserting these into the equation, we calculate the discriminant \(b^2 - 4ac = 36\), which is positive, indicating two distinct real roots.
Therefore, the quadratic formula helps us find these roots, \( x = 2 \) and \( x = -4 \), crucial for solving quadratic inequalities.

Roots of Equation

The roots of a quadratic equation are the values of \( x \) that make the equation equal to zero. These roots are significant as they help break down the real number line into segments, enabling us to analyze the sign variations of the inequality.
When we solved the equation \( x^2 + 2x - 8 = 0 \) from our exercise, we discovered the roots \( x = 2 \) and \( x = -4 \). These roots split the number line into the intervals \((-\infty, -4)\), \((-4, 2)\), and \((2, \infty)\).
In the context of solving inequalities, identifying where the quadratic expression shifts from positive to negative (or vice versa) is key. The roots themselves usually indicate a shift in the sign. This insight guides the steps needed to check which intervals satisfy the inequality at hand.

Inequality Solution

Solving a quadratic inequality involves not just finding roots but also understanding where the expression is positive or negative. Here, the inequality \(-x^2 - 2x + 8 > 0\) translates to finding where the expression is above the \( x \)-axis.
After finding the roots \( x = 2 \) and \( x = -4 \), we categorized our number line into three regions: \((-\infty, -4)\), \((-4, 2)\), and \((2, \infty)\). Through substituting test points from each region into the inequality, we concluded:

• The inequality holds true for the interval \((-4, 2)\).
• It does not hold for \((-\infty, -4)\) or \((2, \infty)\).

This technique lets us observe the behavior of the inequality across different regions, revealing where the solution lies.

Sign Analysis

Sign analysis is pivotal in understanding a quadratic inequality's solution by inspecting the sign changes across its roots. By examining different test points within the intervals defined by the roots, we can determine where the quadratic expression satisfies the inequality.
Here's how you do it:

• Choose test points from each of the intervals created by the roots. For our exercise, we picked \(x = -5\) for \((-\infty, -4)\), \(x = 0\) for \((-4, 2)\), and \(x = 3\) for \((2, \infty)\).
• Substitute these numbers into the original inequality. This provides insight into the sign of the quadratic expression in that interval.

For this problem, only the interval \((-4, 2)\) met the condition \(-x^2 - 2x + 8 > 0\), confirming our solution. Hence, conducting a sign analysis is a decisive factor in solving quadratic inequalities effectively.