Question

Find the zeros of the function. \(f(x)=7 x^2+42\)

Short answer

The given function \(f(x) = 7x^2 + 42\) has no zeros in the set of real numbers.

Step-by-step solution

Identify the coefficients

In the equation \(f(x) = 7x^2 + 42\), we identify that the coefficient of \(x^2\) i.e., \(a = 7\), and the constant term \(c = 42\). Note that there is no term involving \(x\), so the coefficient of \(x\), \(b = 0\).

Formulate the quadratic equation

We are tasked with finding the zeros of the function. This means we are setting \(f(x)=0\) and solving for \(x\). This gives us the equation: \(0 = 7x^2 + 42\).

Solve for x

First divide all terms by 7 to simplify: \(0 = x^2 + 6\). Then, subtract 6 from both sides: \(x^2 = -6\). Here, square roots of negative number do not exist in real numbers, hence the quadratic equation has no zero, i.e., there are no values of \(x\) for which \(f(x)\) equals zero.

Key concepts

Zeros of a Function

When we talk about the zeros of a function, we're looking for the values of \(x\) that make the function equal to zero. Essentially, these are the points where the graph of the function crosses the x-axis. Let's consider the function \(f(x) = 7x^2 + 42\). To find its zeros, we set it equal to zero: \(f(x) = 0\). Now, our objective is to solve for \(x\) in the equation \(7x^2 + 42 = 0\).

In this equation, you notice that there’s no term involving \(x\) by itself – it's a pure quadratic equation. By isolating \(x^2\), we'll find that it equals a negative value, which leads us to explore numbers beyond the real number system.

Complex Numbers

When we encounter square roots of negative numbers in mathematics, we're entering the world of complex numbers. These numbers are an extension of the regular numbers you use every day. They’re handy because they allow us to solve equations that have no solutions among real numbers. Consider the equation \(x^2 = -6\).

Typically, square roots of negative numbers don't exist in the realm of real numbers. But using complex numbers, we overcome this challenge. The imaginary unit \(i\) is defined as the square root of \(-1\). This means \(\sqrt{-6}\) can be expressed as \(i\sqrt{6}\). Therefore, our solutions for \(x\) are \(x = \pm i\sqrt{6}\).

Understanding complex numbers allows us to expand our ability to solve quadratic equations, ensuring no equation is left unsolvable.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions or "zeros" of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is written as:

• \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\)

By substituting the values of \(a\), \(b\), and \(c\) from any quadratic equation, we can solve for \(x\).

In our example, the given function is \(f(x) = 7x^2 + 42\), where \(a = 7\), \(b = 0\), and \(c = 42\). When we try to use the quadratic formula here, we plug in these values. The critical part is the discriminant, \(b^2 - 4ac\). For this example, it turns out to be negative, confirming our earlier conclusion that the solutions are complex numbers. Here, \(x\) involves the imaginary unit \(i\).

The quadratic formula doesn't just help find zeros but gives insight into the type of solutions (real or complex) even before solving the equation completely. It’s like a roadmap for figuring out what kind of numbers we’re dealing with in a quadratic equation.