Problem 39 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$x^{2}+8 x+32=0$$

Step-by-Step Solution

Answer

The solution to the equation \(x^{2}+8x+32\) by completing the square method is \(x=-4\pm4i\). It is further confirmed graphically that the roots are not real but imaginary.

Step 1: Rewrite the quadratic equation

Re-arrange the equation in the form \(x^{2}+8x = -32\).

Step 2: Complete the square

To complete the square, take half of the coefficient of \(x\), square it then add it to both sides of the equation. Half of 8 is 4. The square of 4 is 16. Adding 16 to both sides gives \((x^{2}+8x+16) = -32+16\). The left side of the equation can now be written as \((x+4)^2 = -16\).

Step 3: Solve for x

Taking the square root of both sides we have \(x+4= \pm \sqrt{-16}\). As \(\sqrt{-16}\) is a complex number, we get \(x+4 = \pm 4i\). Therefore, \(x = -4 \pm 4i\).

Step 4: Verify the solution graphically

Now, plotting the equation \(x^{2}+8x+32\), it is seen that there are no real roots as the graph doesn't cross the x-axis but it touches the imaginary axis at -4 , verifying our solution.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fascinating mathematical expressions characterized by the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). At the heart of a quadratic equation is the term "quadratic," which implies that the variable \(x\) is raised to the power of two, i.e., squared.
To solve quadratic equations, several methods can be employed. One popular technique is 'completing the square,' which is particularly useful when the equation does not factor neatly. The goal here is to restructure the equation into a perfect square trinomial form, making it easier to solve for \(x\).
Consider the equation \(x^{2} + 8x + 32 = 0\):

• Reorganize it as \(x^{2} + 8x = -32\).
• Then, take half of the coefficient of \(x\), which is 8, halve it to get 4, and square it to get 16.
• Add this square to both sides, transforming it into \((x + 4)^{2} = -16\).

This form allows for solving by taking the square root of both sides. Once in this form, the quadratic equation becomes much more straightforward to solve.

Introduction to Complex Numbers

Complex numbers extend our ability to solve equations beyond the limitations of real numbers. They are especially useful when dealing with quadratic equations like \((x+4)^2 = -16\), which yield no real number solutions.
A complex number is composed of a real part and an imaginary part. The imaginary unit, denoted by \(i\), is defined by the property \(i^2 = -1\).
When solving our equation using complex numbers:

• After completing the square, we find \(x+4 = \pm \sqrt{-16}\).
• The square root of \(-16\) is \(\pm 4i\), utilizing the property of \(i\).

Thus, the solutions to the equation are \(x = -4 + 4i\) and \(x = -4 - 4i\). These reflect the complex nature of solutions for certain quadratic equations. Understanding complex numbers is essential in accepting solutions that do not reside on the real number line.

Graphical Verification

Graphical verification is a powerful tool in mathematics, offering a visual check of algebraic solutions. In the context of a quadratic equation, this involves plotting the equation's function to explore its intersections with the axes.
For the equation \(x^2 + 8x + 32 = 0\), plotting the corresponding function \(y = x^2 + 8x + 32\) provides insight into the nature of its roots:

• The graph of this function is a parabola opening upwards.
• Because the graph does not intersect the x-axis, there are no real roots – affirming that any solutions are not visible in the real number domain.

Instead, the parabola's vertex exists below the x-axis and does not cross it, confirming any intersections occur purely in the complex plane. Graphical verification strengthens the validity of solutions by visually demonstrating the absence or presence of real roots, further concluding the legitimacy of complex solutions such as \(-4 \pm 4i\). This method not only validates solutions but also enhances comprehension through visual means.