Problem 27 - Precalculus : Building Concepts and Connections

Question

Solve the inequality by factoring. $$-x^{2}+2 x-1<0$$

Step-by-Step Solution

Answer

The inequality \(-x^{2}+2 x-1<0\) has no solution.

Step 1: Factorize the quadratic polynomial

Firstly, the quadratic polynomial \(-x^{2}+2 x-1\) is factored to find its roots. It is factored into \(-(x-1)^2\).

Step 2: Find the critical points

The roots or zeros of the equation represent the critical points. Set \(-(x-1)^2 = 0\). The solution for x is \(1\). Thus, \(1\) is the critical point.

Step 3: Determine the intervals

The critical point splits the number line into two intervals: \(-\infty ,1\) and \(1,+\infty \). These intervals will be tested to determine where the function is negative and satisfies the inequality.

Step 4: Test the intervals

Test a value from each interval in the factored equation \(-(x-1)^2\), here, for example 0 and 2 can be used, for \(-\infty ,1\) and \(1,+\infty \) respectively. Substituting these values in, for \(x=0\), \((0-1)^2=1>0\) and for \(x=2\), \((2-1)^2=1>0\). For both intervals, the result is positive, therefore no interval satisfies the inequality.

Step 5: Answer

Based upon the tests, there are no intervals in which the inequality \(-x^{2}+2 x-1<0\) holds true.

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential technique in algebra and helps to simplify quadratic expressions by representing them as the product of their factors. These factors can then be used to find the roots or solutions of the quadratic equation. In the given inequality \(-x^{2}+2x-1<0\), factoring is applied initially to help solve the inequality. This expression can be rewritten as \(- (x-1)^2 \), by identifying \(x-1\) as the repeated factor for the original polynomial. By successfully factoring, we begin to see the equation in its simplest terms.
Understanding how to factor is crucial because it transforms complex expressions into more manageable forms which are easier to work with.

• Identify any common factors that all terms share.
• Recognize patterns such as perfect square trinomials or the difference of squares.
• Rearrange terms and apply the appropriate factoring operations.

Critical Points

Critical points occur where the expression being evaluated changes its sign or its slope becomes zero. These points are often the solutions or roots of the quadratic where it equals zero. In our inequality, setting the factored quadratic \(-(x-1)^2=0\) reveals that the equation only has the root \(x = 1\). This root is our critical point.
The critical point divides the number line into distinct intervals where the value of the quadratic will behave differently. Analyzing behavior at the critical points helps understand how the function changes across the number line, assisting in solving inequalities.

• Found by setting the factored form equal to zero.
• Helps in breaking down intervals to test for sign changes in the inequality.
• Critical for determining where inequalities hold true.

Interval Testing

Interval testing involves plugging different values from specific intervals back into the factored equation to determine whether they satisfy the inequality. Since our inequality is separated into intervals by \(x = 1\), we have two main intervals to test: \(-\infty, 1\) and \(1, \infty\). By substituting values like 0 from \(-\infty, 1\) and 2 from \(1, \infty\),we can observe how the inequality behaves.
For instance, at \(x = 0\), \(-(0-1)^2\) results in a positive outcome, indicating it does not satisfy the inequality \(-x^2+2x-1 < 0\). Moreover, when checking \(x = 2\), \(-(2-1)^2\) also results in a positive number, further confirming this behavior for the whole interval.

• Exercises involve choosing test values well within the interval.
• Each test value will determine if the inequality is satisfied.
• Testing both sides of a critical point helps confirm solutions or lack thereof.

Inequality Solving Methods

Solving inequalities using factoring and interval testing requires a grasp of various techniques that extend beyond simply solving for \(x\). We navigate quadratic inequalities by transforming them into a product of factors, finding critical points, and testing intervals. Here, methods such as these are combined to deduce where the quadratic expression is less than zero.
Ultimately, when all interval tests resulted in a positive inequality in our specific exercise, it displays how no value of \(x\) actually satisfies this inequality \(-x^2 + 2x - 1 < 0\). Other common methods involve graphing or completing the square, but in this example, the emphasis remains on algebraic techniques.

• Combines factoring, interval testing, and critical points for comprehensive understanding.
• Crucial for understanding where solutions occur in inequalities.
• Serves as a handy toolset to approach various forms of inequalities efficiently.