Question

Find the zeros of the function. \(f(x)=-\frac{1}{5} x^2-10\)

Short answer

The zeros of the function are \(x = 5i\sqrt{2}\) and \(x = -5i\sqrt{2}\)

Step-by-step solution

Write the function in standard form

First, set the function equal to zero in order to find its roots. This gives the equation \(-\frac{1}{5} x^2-10=0\). This is already in the standard form for a quadratic equation, \(ax^2+bx+c=0\), with \(a=-\frac{1}{5}\), \(b=0\), and \(c=-10\)

Solve for x

The quadratic formula can be used to find the roots of the equation, but in this case, the equation can be easily rearranged to find \(x\). Multiply through by -5 to get rid of the fraction: \(x^2 + 50 = 0\). Then, subtract 50 from both sides: \(x^2 = -50\)

Extract the square roots

Take the square root of both sides of the equation. This yields two solutions (since \(\sqrt{-1} = i\)): \(x = \sqrt{-50} = 5i\sqrt{2}\) and \(x = -\sqrt{-50} = -5i\sqrt{2}\)

Key concepts

Zero of a function

The "zero of a function" refers to the input values (often denoted as "x") that make the function equal to zero. In other words, if you plug these x-values into the function, the output will be zero. These are also known as roots or solutions of the equation.
To find the zeros of a function like a quadratic function, you typically set the function equal to zero and solve for x. For example, given the function \[f(x) = -\frac{1}{5}x^2 - 10 \]you start by setting it equal to zero:\[-\frac{1}{5}x^2 - 10 = 0\]This equation is in the standard form for a quadratic equation, ready to solve for its zeros.

The zero of a function is a vital concept in algebra as it reveals specific points where the graph of the function intersects the x-axis.

Quadratic Equation

A quadratic equation is any equation that can be rearranged in standard form as:\[ax^2 + bx + c = 0\]where "a", "b", and "c" are constants, with "a" not equal to zero. This is crucial because if "a" were zero, the equation would not be quadratic, but rather linear.
Quadratic equations can be solved using several methods:

• Factoring, when applicable and straightforward.
• Completing the square, useful especially when solving derived equations.
• Using the quadratic formula, \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] which is applicable to all forms of quadratic equations.

In the initial exercise, rearranging involved multiplying by -5 to clear the fraction, yielding:\[x^2 + 50 = 0\]This equation is quadratic as it follows the form "ax^2 + c = 0", and we can proceed to solve for x.

Complex Numbers

Complex numbers extend the idea of one-dimensional real numbers to the two-dimensional complex plane using an imaginary unit, i (where \i^2 = -1). This makes it possible to solve equations like \[x^2 = -50\]which would be undefined in the real number system.

When we take the square root of negative numbers, complex numbers come into play. In our problem, \[x^2 = -50 \]turns into two solutions:\[x = 5i\sqrt{2} \quad \text{and} \quad x = -5i\sqrt{2}\]These solutions illustrate how complex numbers are used to find roots of quadratic equations that lack real solutions. This is especially true for equations where the discriminant \(b^2 - 4ac\) is negative, indicating that the parabola represented by the quadratic function does not touch the x-axis.
Understanding complex numbers is essential as they open the door to solving equations that extend beyond the reach of purely real numbers.