Question

Find the zeros of the function. \(h(x)=2 x^2+72\)

Short answer

The zeros of the function \(h(x) = 2x^2 + 72\) are ±6i.

Step-by-step solution

Set the equation to zero

The first step to find the zeros of \(h(x) = 2x^2 + 72\) is setting the equation equal to zero. This gives us \(2x^2 + 72 = 0\).

Rearrange the equation to solve for \(x^2\)

Subtract 72 from both sides of the equation, resulting in \(2x^2 = -72\). Then, divide both sides by 2 to isolate \(x^2\). This yields \(x^2 = -36\).

Find the square root

The solutions for \(x\) will be the positive and negative square roots of -36. But the square of a negative number is not defined in the real number system. Therefore, we introduce the imaginary unit \(i\), where \(i^2 = -1\). This gives us \(x = \sqrt{-36} = \pm 6i\).

Key concepts

Quadratic equations

Quadratic equations are fundamental in algebra and often take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. The solutions to a quadratic equation represent the values of \(x\) that make the equation equal zero. These solutions are also referred to as the "roots" or "zeros" of the equation.

In practice, quadratic equations can be solved using a variety of methods, such as factoring, completing the square, or using the quadratic formula. The nature of the solutions—whether they are real or complex—depends on the discriminant, given by \(b^2 - 4ac\).

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution.
• If the discriminant is negative, the solutions are complex or imaginary numbers.

In the given exercise, the equation \(2x^2 + 72 = 0\) is an example of a quadratic equation; however, for the solution to exist, we needed to handle the imaginary unit since its discriminant is negative.

Imaginary unit

The imaginary unit is a mathematical concept used to extend the real number system to handle the square roots of negative numbers. It is denoted by \(i\), and it is defined as \(i^2 = -1\). This seems a bit mysterious at first, but it becomes clearer when you think of \(i\) as a tool that allows us to work with challenges that don't fit into the ordinary world of real numbers.

In the earlier problem, when finding the zeros of \(2x^2 + 72\), we encountered \(-36\). To take the square root of \(-36\), we used the imaginary unit: \( \sqrt{-36} = \sqrt{36} \times \sqrt{-1} = 6i \), giving us solutions \(x = 6i\) and \(x = -6i\).

Working with \(i\) is essential in fields like electrical engineering, where complex numbers provide a complete framework for frequencies and signals. It's a fascinating area where algebra meets the real world in a very tangible way.

Square roots

The square root of a number \(a\) is the value \(x\) such that \(x^2 = a\). Normally, when \(a\) is positive, there are two solutions, \(\sqrt{a}\) and \(-\sqrt{a}\), as both satisfy the original equation when squared.

However, things get intriguing when \(a\) is negative. In real numbers, the square root of a negative number is undefined because there's no real number squared that results in a negative value. This is where the imaginary unit \(i\) becomes crucial, allowing us to define the square roots of negative numbers.

• For example, \(\sqrt{-36}\) is not possible with real numbers. Using \(i\), it becomes \(6i\), since \( \sqrt{-36} = \sqrt{36 \times (-1)} = 6i \).
• This implies complex solutions for equations like \(2x^2 + 72 = 0\), where only through the power of \(i\) can we find the roots.

Understanding square roots in both their real and complex forms is essential for solving equations and working with complex numbers in various mathematical and engineering applications.