Problem 60 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$x^{2}+16=-5 x$$

Step-by-Step Solution

Answer

The solution to the equation \(x^2 + 5x + 16 = 0\) are complex numbers: \( x = -2.5 \pm 3.125i \)

Step 1: Re-write in standard form

To solve a quadratic equation using the Quadratic Formula, it must first be written in standard form. Standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). Rearranging the equation \( x^2 + 16 = -5x \) to match this form gives \(x^2 + 5x + 16 = 0\). Therefore, for this equation, \( a = 1, b = 5, c = 16 \).

Step 2: Use Quadratic formula

The Quadratic Formula is \( x = \[-b \pm sqrt(b^{2} - 4ac) \] / \[2a\]\). Substituting in the values from step 1, \( b = 5 \), \( a = 1 \), \( c = 16 \), into the formula give: \(x = \[-(5) \pm sqrt((5)^{2} - 4*1*16)\] / \[2*1\]\). Thus we have \(x = \[-5 \pm sqrt(25 - 64)\] / \[2\]\)

Step 3: Solve for x

Solve the equation \[x = -5 \pm sqrt(-39) / 2\]. Notice that the term under the square root is negative, which means there are no real roots to this equation. Thus, we have two complex solutions: \( x = -2.5 \pm 3.125i \).

Step 4: Verify solutions graphically

Graph the equation \( x^2 + 5x + 16 \) using a graphing utility. Since the roots are complex, there are no points where the graph crosses the x-axis. This confirms the solutions found algebraically.

Key Concepts

Complex Solutions

When solving quadratic equations, sometimes we encounter complex solutions. This happens when the quadratic equation's discriminant, which is found under the square root in the quadratic formula, is negative. The discriminant for a quadratic equation of the form \( ax^2 + bx + c = 0 \) is calculated as \( b^2 - 4ac \).

In our problem, substituting the values \( a = 1 \), \( b = 5 \), and \( c = 16 \) into the discriminant gives us \( 25 - 64 = -39 \). Since -39 is negative, the solutions aren't real numbers. Instead, they are complex numbers, which include the imaginary unit \( i \), where \( i^2 = -1 \).

For complex solutions, the roots of the equation are written as \( -2.5 \pm 3.125i \). The term \( 3.125i \) originates from simplifying the square root of -39. Complex solutions indicate that the graph of the equation does not intersect the x-axis, as no real solutions exist.

Graphing Utility

A graphing utility is an instrumental tool in verifying the nature of solutions for a quadratic equation. By inputting the quadratic equation into a graphing utility, you can visually confirm whether the solutions calculated are correct.

For the equation \( x^2 + 5x + 16 \), when plotted on a graphing utility, the graph is a parabola that does not touch or cross the x-axis. This visual confirmation supports the algebraic finding that the solutions to the equation are complex, with no real roots.

Using a graphing utility is beneficial because:

• It offers a quick way to check the nature of the roots — whether they are real, repeated, or complex.
• It provides a clear, visual representation of the function.
• It helps in understanding how the solutions relate to the graph's shape and position.

Even though graphing utilities do not solve the equation algebraically, they are invaluable for confirming your work and enhancing comprehension.

Standard Form

The standard form of a quadratic equation is essential for applying the Quadratic Formula. It is written as \( ax^2 + bx + c = 0 \). Transforming an equation into this format is crucial before using the Quadratic Formula to solve it.

To achieve the standard form from an equation like \( x^2 + 16 = -5x \), rearrange terms to align with \( ax^2 + bx + c = 0 \). For this specific exercise, adding \( 5x \) to both sides results in \( x^2 + 5x + 16 = 0 \).

In the context of quadratic equations:

• \( a \) is the coefficient of \( x^2 \).
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

Identifying these coefficients allows for straightforward substitution into the Quadratic Formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), facilitating the solving process. Understanding standard form simplifies the use of various methods for solving quadratic equations and plays a foundational role in algebra.