Problem 26 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ -x^{2}-6 x+7 \leq 0 $$

Step-by-Step Solution

Answer

The solution is \(-7 \leq x \leq 1\).

Step 1: Rewrite the Inequality in Standard Form

First, let's analyze the inequality \[-x^2 - 6x + 7 \leq 0\]. Realize that this is a quadratic inequality. In standard form, it stays the same because it's already set equal to zero on one side: \[-x^2 - 6x + 7 \leq 0\].

Step 2: Determine the Parabola's Direction

The quadratic term \(-x^2\) has a negative coefficient, which indicates that the parabola opens downwards. This will help us determine whether the solutions are between the roots or outside them.

Step 3: Find the Roots of the Quadratic Equation

Set the inequality as an equation to find the roots: \[-x^2 - 6x + 7 = 0\]. Multiply through by -1 to simplify solving: \[x^2 + 6x - 7 = 0\]. Use the quadratic formula, \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\], with \(a = 1\), \(b = 6\), and \(c = -7\). Calculate:\[x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} = \frac{-6 \pm \sqrt{36 + 28}}{2} = \frac{-6 \pm \sqrt{64}}{2} = \frac{-6 \pm 8}{2}\].The solutions are \(x = 1\) and \(x = -7\) which are the roots of the equation.

Step 4: Determine the Interval for the Inequality

Since the parabola opens downwards, the inequality \(-x^2 - 6x + 7 \leq 0 \) has solutions between the roots \(-7\) and \(1\). Hence, the solution set where the inequality holds is \(-7 \leq x \leq 1\).

Step 5: Confirm Algebraically and with a Table or Graph

Verify by substituting values from the intervals and check the signs of the expressions either side of the intervals or use a graphing tool. A table can also be created: | \(x\) | \(-x^2 - 6x + 7\) ||---------|------------------|| -8 | 15 || -7 | 0 || 0 | 7 || 1 | 0 || 2 | -9 |This confirms that the expression is zero or less between \(-7\) and \(1\).

Key Concepts

Quadratic Equation

Quadratic equations form the backbone of many mathematical concepts. A quadratic equation is a second-degree polynomial equation in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). This polynomial's degree, being two, makes it quadratic, and thus it involves the square of the variable \(x\).
In the exercise we analyzed, the given inequality can be viewed as a quadratic equation by equating it to zero:

• The original inequality: \(-x^2 - 6x + 7 \leq 0\).
• Rewritten for solving roots: \(-x^2 - 6x + 7 = 0\).

Understanding the structure of a quadratic equation helps in identifying solutions, whether through algebra or graphing techniques.

Parabola

The visual representation of a quadratic equation on a graph is known as a parabola. Parabolas are symmetrical, U-shaped curves which can open upwards or downwards depending on the sign of the coefficient of the \(x^2\) term.
The formula used was \(-x^2 - 6x + 7\), indicating a downward-opening parabola due to the negative \(x^2\) term. This direction is crucial as it guides us in determining where the solutions to the inequality lie. For a downward-opening parabola:

• The solutions to the inequality \(-x^2 - 6x + 7 \leq 0\) are found between the vertex and the points where the parabola intersects the x-axis.

This understanding aids in analyzing not just the roots, but also the overall behavior of the function on a graph.

Inequality Solutions

Solving quadratic inequalities involves identifying the range of values for which the inequality holds true. Unlike quadratic equations, inequalities utilize symbols such as \(\leq\) or \(<\) rather than \(=\).
To handle the inequality \(-x^2 - 6x + 7 \leq 0\), we first found the roots by solving the quadratic equation \(-x^2 - 6x + 7 = 0\). Analyzing the nature of the graph, we concluded that the values for which the inequality holds were between \(-7\) and \(1\).
Effectively solving inequalities requires different methods such as:

• Algebraic manipulation to find critical points.
• Graphical analysis to visualize the solution set.
• Tables or sign charts to verify solution intervals.

All these approaches ensure that we capture the correct set of solutions.

Quadratic Roots

The roots of a quadratic equation are values of \(x\) that make the equation equal to zero. They are the points where the parabola crosses the x-axis. Finding roots is essential in solving quadratic inequalities.
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we computed the roots for the equation \(-x^2 - 6x + 7 = 0\):

• This becomes \(x^2 + 6x - 7 = 0\) after multiplying through by \(-1\).
• The roots found were \(x = 1\) and \(x = -7\).

Recognizing the roots enabled us to determine the region of the number line for which the inequality stands true. Understanding these roots helps to compare with an inequality to pinpoint exact solution intervals.