Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the inequality and graph the solution on the real number line. Use a graphing utility to verify your solution graphically. $$x^{2}+3 x+8>0$$

Step-by-Step Solution

Answer

The solution to the inequality \(x^{2}+3x+8>0\) is all real numbers.

Step 1: Simplifying the inequality

The inequality can be kept as is because it's already simplified. So, the given inequality is \(x^{2}+3x+8>0\)

Step 2: Finding the roots

For the roots, we would set the equation part \(x^{2}+3x+8=0\). However, calculating the discriminant \(b^{2}-4ac\) which is \(3^{2}-4*1*8\) gives -23. Since this value is negative, there are no real roots. Therefore, this quadratic doesn't intersect with x-axis.

Step 3: Finding the sign of Quadratic Function

Analyze the sign of the leading coefficient of the quadratic function to determine the behaviour of the graph. The coefficient of \(x^{2}\) is positive, meaning the parabola opens upwards.

Step 4: Graphing and verifying with a graphing utility

Graph the parabola \(y=x^{2}+3x+8\). It would be a parabola opening upwards and not intersecting x-axis at all. Thus the given quadratic is always positive for all real x, i.e., \(x^{2}+3x+8>0\) is true for all real x. You can verify these conclusions using a graphing utility.

Key Concepts

Graphing Quadratic Functions

Understanding how to graph a quadratic function is crucial when working with quadratic inequalities. A quadratic function can be expressed in the form of \( y=ax^2+bx+c \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero.

To graph a quadratic function, start by identifying the vertex—the highest or lowest point of the parabola, depending on the sign of \( a \). If \( a > 0 \), the parabola opens upwards, and if \( a < 0 \), it opens downwards. The vertex can be found using the formula \( x = -\frac{b}{2a} \), and then substituting this \( x \) value back into the original equation to find the \( y \) coordinate.

Next, determine if there are any real roots or x-intercepts by setting the quadratic equation to zero and solving for \( x \). The axis of symmetry, which is a vertical line that passes through the vertex, can also be determined and helps ensure the parabola is symmetric. Once you have these elements, you can sketch the parabola noting that it is a smooth curve.

Discriminant of Quadratic Equations

The discriminant is a valuable tool for analyzing quadratic equations without actually solving them. It is part of the quadratic formula, \( \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), and is the expression under the square root, \( b^2-4ac \).

The value of the discriminant can tell us about the number and type of roots a quadratic equation has. If the discriminant is positive, there are two distinct real roots. If it is zero, there is exactly one real root, which means the parabola touches the x-axis just once. And if the discriminant is negative, as in our exercise, there are no real roots; the parabola does not intersect the x-axis at all. Understanding the discriminant helps us anticipate the shape and position of the parabola on the graph without lengthy calculations.

Quadratic Inequalities

Quadratic inequalities, such as the one in our example \( x^2+3x+8>0 \), can be solved easily once we have the graph of the associated quadratic function. The inequality asks for which \( x \) values the function outputs are positive or negative.

When we have a quadratic inequality, we would typically look for the roots to divide the number line into intervals where we can test the sign of the function. However, in this case, since the discriminant revealed no real roots, we know the parabola never crosses the x-axis and retains the same sign everywhere.

Furthermore, since the coefficient of \( x^2 \) is positive, we know the parabola opens upwards, and thus the y-values (or the value of the quadratic function) are always positive. Therefore, the inequality \( x^2+3x+8>0 \) is true for all real \( x \). By combining knowledge of quadratic functions, the discriminant, and the behavior of inequalities, we can solve and graph quadratic inequalities effectively.