Problem 37 - Algebra 2

Question

Solve each inequality using a graph, a table, or algebraically. $$ x^{2}+12 x<-36 $$

Step-by-Step Solution

Answer

No solution exists, as the quadratic never falls below zero.

Step 1: Set the Inequality as a Quadratic Equation

First, we need to transform the inequality into a quadratic equation by replacing the inequality sign with an equality. So we have: \( x^{2} + 12x + 36 = 0 \). This will help us to find the boundary values.

Step 2: Solve the Quadratic Equation

We solve \( x^{2} + 12x + 36 = 0 \) to find the roots. First, we recognize this as a perfect square trinomial: \( (x + 6)^2 = 0 \). Therefore, the root is \( x = -6 \). This tells us that the graph of the quadratic touches the x-axis at this point.

Step 3: Determine the Nature of the Inequality

The inequality \( x^{2} + 12x + 36 < 0 \) implies the parabola is below the x-axis. Since \( (x + 6)^2 \) is only zero at \( x = -6 \) and positive otherwise, there is no solution where \( x^{2} + 12x + 36 < 0 \).

Step 4: Check the Solution

Re-evaluate to ensure the steps are correct. Observing the nature of \( (x + 6)^2 \), it's evident that \( x^{2} + 12x + 36 \) cannot be less than zero, confirming no solution exists for \( x^{2} + 12x + 36 < 0 \).

Key Concepts

Graphing Inequalities

Exploring quadratic inequalities through graphing can make them more approachable. When graphing a quadratic inequality like \(x^{2} + 12x < -36\), it involves visualizing where the quadratic expression is above or below the x-axis.

The inequality is first converted into an equation format: \(x^{2} + 12x + 36 = 0\). By plotting the graph of \(y = x^{2} + 12x + 36\), we get a parabola. The orientation of this parabola is upwards since the coefficient of \(x^2\) is positive.

• The vertex represents the maximum or minimum point of the parabola.
• The x-intercepts are the points where the parabola crosses the x-axis.

For this quadratic, we find a perfect square trinomial, \((x + 6)^2 = 0\), showing the parabola touches the x-axis only at \(x = -6\). Since all values of \((x + 6)^2\) are zero or positive, and it symbolizes the whole construction being above the axis or touching it, there are no x-values making the inequality \(x^{2} + 12x + 36 < 0\) true. Hence, the graph confirms there are no solutions below the x-axis.

Quadratic Equation Roots

Finding the roots of a quadratic equation is essential when analyzing inequalities. Roots indicate where the parabola crosses the x-axis, marking crucial points for solving the inequality.

The inequality \(x^{2} + 12x < -36\) translates to \(x^{2} + 12x + 36 = 0\). Identifying the roots involves solving \(x^{2} + 12x + 36 = 0\).

• This quadratic is recognized as a perfect square trinomial, \((x + 6)^2\).
• Simplifying \((x + 6)^2 = 0\) yields a double root: \(x = -6\).

This means the parabola touches the x-axis at \(x = -6\) but doesn’t cross it. In terms of the inequality \(x^{2} + 12x + 36 < 0\), we seek where the parabola dips below, which it doesn't, so there is no real solution where it satisfies the inequality to be less than zero. The roots often guide the intervals for testing within inequalities, showing where the parabola changes its direction relative to the x-axis.

Perfect Square Trinomial

A perfect square trinomial is a quadratic that can be factored into identical binomials squared. Recognizing such trinomials simplifies solving processes for quadratic inequalities.

In the exercise \(x^{2} + 12x + 36 < 0\), converting to an equation we get \(x^{2} + 12x + 36 = 0\). We identify \((x + 6)^2 = 0\) as a perfect square trinomial. This means:

• The quadratic is the square of a binomial \((x + 6)\).
• It expands to \(x^{2} + 12x + 36\).

Understanding this helps us instantly see the impact of its roots. The root \(x = -6\) is a point where the parabola does not cross but just touches the x-axis. The property of perfect squares implies all output values are non-negative unless \(x = -6\), meaning this trinomial always results in zero or positive values. Consequently, the expression cannot satisfy being less than zero, simplifying the process of resolving such mathematical inequality scenarios.