Problem 38 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ -x^{2}+14 x-49 \geq 0 $$

Step-by-Step Solution

Answer

The solution is \(x = 7\).

Step 1: Express the Inequality as an Equation

The first step is to set the inequality \(-x^{2}+14 x-49 \geq 0\) equal to zero to find the critical points. This can be rewritten as:\[-x^{2}+14x-49 = 0\].

Step 2: Simplify the Quadratic Equation

Multiply the entire equation by \(-1\) to make it easier to factor:\[x^{2} - 14x + 49 = 0\].

Step 3: Factor the Quadratic Equation

Factor the quadratic equation. Notice that the equation is a perfect square trinomial:\[(x-7)^2 = 0\].

Step 4: Solve for Critical Points

Solve the factored equation:\[(x-7)^2 = 0\].This implies \(x-7 = 0\).Thus, \(x = 7\) is the only critical point.

Step 5: Test Intervals Around the Critical Point

Choose test points around the critical point \(x = 7\), such as \(x = 6\) and \(x = 8\), to determine where the inequality is true: - For \(x = 6\): \(-6^2 + 14 \cdot 6 - 49 = -1\) (this is negative, so \(6\) is not a solution). - For \(x = 8\): \(-8^2 + 14 \cdot 8 - 49 = -1\) (this is negative, so \(8\) is not a solution).This shows that the inequality is not satisfied anywhere except at \(x = 7\).

Step 6: Conclusion of Solution

The solution to the inequality \(-x^{2}+14x-49 \geq 0\) is simply the point \(x = 7\).

Key Concepts

Critical Points

In the context of quadratic inequalities, critical points play a significant role. They are the points where the equation is set to zero. This is because they reveal where the graph of the quadratic equation touches or crosses the x-axis. To find them, set the quadratic expression equal to zero and solve for the variable. In our example, the equation is:

• Find the solutions of the equation \(x^2 - 14x + 49 = 0\).

The solutions to this equation, if any, are the critical points. These points mark where the inequality might change from true to false or vice versa. In our problem, solving \(x^2 - 14x + 49 = 0\) gives the critical point \(x = 7\).
The critical point helps us understand the range of values that satisfy the inequality.

Factoring Quadratics

Factoring quadratics is a method used to simplify quadratic equations, making them easier to solve or analyze. It involves rewriting the quadratic equation as a product of two binomial expressions. Our quadratic expression \(x^2 - 14x + 49\) is a perfect candidate for factoring.
For perfect square trinomials like \(x^2 - 14x + 49\), the quadratic can often be factored into two identical binomials. In our case:

• Identify that \(x^2 - 14x + 49\) can be factored as \((x - 7)(x - 7)\) or \( (x - 7)^2\).

This factorization helps simplify determining the critical points and understanding the nature of the inequality we're dealing with.

Test Intervals

After determining the critical point(s), it's essential to test the intervals created by these points. This tells us where the inequality holds true. Since we have a single critical point at \(x = 7\), the number line splits into two intervals: \(x < 7\) and \(x > 7\).

• To test, choose any value from each interval. For example, use \(x = 6\) and \(x = 8\).
• Plug these test values into the inequality, solving to see if it holds true.
• For \(x = 6\) and \(x = 8\), both cases return negative results, meaning the inequality does not hold in these intervals.

As a result, we conclude the inequality only holds true at the critical point itself, \(x = 7\).

Perfect Square Trinomials

Perfect square trinomials are special types of quadratic expressions that can be expressed as the square of a binomial. Recognizing and factoring these correctly is a valuable skill. A perfect square trinomial fits the pattern \(a^2 - 2ab + b^2 = (a - b)^2\).
For the quadratic given in the exercise, \(x^2 - 14x + 49\), we see:

• This completes to the trinomial \((x - 7)^2\).
• Each part of the expression contributes to the perfect square pattern: \(x^2 = a^2\), \(-14x = -2ab\), and \(+49 = b^2\).

By recognizing this pattern, we factor the equation into \( (x - 7)^2 = 0\), leading us directly to the critical point. This simplifies solving the inequality, highlighting the utility of identifying perfect square trinomials.