Question

Graphical Reasoning Solve \(x^{4}+5 x^{2}+4=0\). Then use a graphing utility to graph \(y=x^{4}+5 x^{2}+4\) What is the connection between the solutions you found and the intercepts of the graph?

Short answer

The solutions to the equation correlate with the x-intercepts of the graph. However, in this case the equation has no real solutions, which corresponds to the lack of x-intercepts on the graph of the function.

Step-by-step solution

Simplify the equation

We notice that \(x^{4}+5x^{2}+4=0\) is a quadratic equation in terms of \(x^{2}\). Such equation can be rewritten in a more familiar form by substituting \(x^{2}=p\). Then our equation becomes \(p^{2}+5p+4=0\).

Solve the quadratic equation

Next step is to solve this quadratic equation by using the quadratic formula \(p=(-b±\sqrt{b^{2}-4ac})/(2a)\). In our case \(a=1\), \(b=5\) and \(c=4\). Substituting these values into the formula, we get \(p=(-5±\sqrt{25-16})/2\), which simplifies to \(p=-1\) or \(p=-4\). Remember that since we substituted \(x^{2}=p\), these are values of \(x^{2}\), not \(x\). Thus, \(x^{2}=-1\) and \(x^{2}=-4\) are original solutions. But they do not have real number solutions, so the original quartic equation has no real solutions.

Analyze the graph

We graph \(y=x^{4}+5x^{2}+4\). Since there are no real solutions to the equation, there will be no x-intercepts on the graph.

Key concepts

Quadratic Formula

Understanding the quadratic formula is essential when dealing with quadratic equations, which are polynomials of the second degree. The general form of a quadratic equation is given by \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}} \), is derived from completing the square in the general equation and provides a straightforward method to find the roots of any quadratic equation.

When applying the formula, one computes the discriminant, \( \Delta = b^2-4ac \), which is the part under the square root. Depending on the value of the discriminant, the equation can have:

• Two distinct real solutions if \( \Delta > 0 \)
• One real solution if \( \Delta = 0 \)
• No real solutions (but two complex solutions) if \( \Delta < 0 \)

As shown in the exercise, using the quadratic formula to solve for \( p \) swiftly facilitates the process of finding potential solutions for \( x \) when dealing with quartic equations in disguised quadratic form.

Graphical Analysis

Graphical analysis is a powerful tool for understanding the behavior of equations by plotting them on a coordinate plane. The graph of a quadratic equation \( y = ax^2 + bx + c \) is a parabola, and its intersections with the x-axis correspond to the roots of the equation.

For higher-degree polynomials, such as the quartic equations in this exercise, graphing can still be utilized to discern important features like symmetry, extrema, and intercepts. A significant point is that the x-intercepts of the graph are indeed the real roots of the equation. However, when there are no real roots, as in the given exercise, the graph of the equation does not cross the x-axis. Instead, it might merely touch or stay above/below the axis entirely, indicating the presence of complex roots or no real solutions at all.

Quartic Equations

Quartic equations, or fourth-degree polynomials, can be more complex to solve than quadratic equations. The general form of a quartic equation is \( ax^4 + bx^3 + cx^2 + dx + e = 0 \), and they can exhibit a range of up to four real roots or a combination of real and complex roots. One method to solve certain quartic equations is to transform them into quadratic form, as was done in the exercise by setting \( x^2 = p \), thus reducing the original quartic equation to a quadratic equation that is more manageable.

This transformation is particularly effective when the equation lacks an \( x^3 \) or \( x \) term, as it becomes symmetrical concerning \( x^2 \), allowing for a simpler approach to finding solutions. While the quartic equation in the exercise looks daunting, recognizing and applying this substitution technique can turn a challenging problem into an achievable one.

Graphing Utility

A graphing utility is an indispensable tool for students and mathematicians alike to visualize and analyze complex mathematical equations. These utilities allow for the plotting of functions, examination of curves, and identification of key points such as intercepts and turns. They can range from simple online graphers to sophisticated software programs or calculators.

Using a graphing utility, one can observe the shape and behavior of the polynomial's graph without manual plotting, which can be especially helpful for higher-degree polynomials. When working through problems like our example, graphing utilities can confirm the absence of real solutions by showing that the curve does not intersect the x-axis. It is important to note that while a graphing utility can suggest the number and nature of solutions, it is usually necessary to use algebraic methods like the quadratic formula to find the exact values of these solutions.