Question

Find the zeros of the function. \(r(x)=-\frac{1}{2} x^2-24\)

Short answer

The zeros of the function \(r(x)=-\frac{1}{2} x^2-24\) are \(x = \pm \sqrt{48}\)

Step-by-step solution

Set the Function Equal to Zero

The first step is to set the function equal to zero. So, \( -\frac{1}{2} x^2-24 = 0\).

Rearrange the Equation

The next step is to rearrange the equation in order to isolate \(x^2\). This gives \(x^2 = -2*(-24)\).

Solve for \(x^2\)

Solve \(x^2 = -2*(-24)\) to find \(x^2 = 48\).

Solve for \(x\)

The last step is to take the square root of both sides of the equation, remembering to include the plus/minus sign. So, \(x = \pm \sqrt{48}\).

Key concepts

Setting Equations to Zero

When finding the zeros of a quadratic function, the initial step involves setting the entire equation equal to zero. This is crucial because the zeros, or roots, of the function are the points where the graph of the function crosses or touches the x-axis. By convention, these points occur when the value of the function is zero. For instance, in the equation \(r(x)=-\frac{1}{2} x^2-24\), we start by rewriting it as \( -\frac{1}{2} x^2-24 = 0\). This action essentially 'prepares' the equation for the methods we'll use to isolate variables and solve it.

Isolating Variables

The next phase in the process is isolating the variable, which involves manipulating the equation to express the variable we're solving for (in this case, \(x\)) on one side by itself. To achieve that, we move terms across the equals sign, changing their signs if necessary, and performing operations like addition, subtraction, multiplication, or division. In our particular problem, rearranging the equation \( -\frac{1}{2} x^2-24 = 0\) leads us to \(x^2 = -2*(-24)\), isolating the \(x^2\) term. By isolating \(x\), we can now proceed to solve for it more directly.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. After isolating the variable, the next critical step is to actually solve for it. In the context of quadratic equations, which are of the form \(ax^2 + bx + c = 0\), our objective is to find the value(s) of \(x\) that satisfy the equation. There are various methods for solving these types of equations, including factoring, completing the square, and using the quadratic formula. In our example, we have \(x^2 = 48\), a simplified form where we can directly apply the next concept: taking square roots.

Square Roots

The final step in finding the zeros of the given quadratic function involves taking the square root of both sides of the equation. The square root operation essentially asks the question, 'What number, when multiplied by itself, gives the original number?' In the equation \(x^2 = 48\), applying the square root to both sides gives us \(x = \pm \sqrt{48}\). It's essential to remember the plus/minus (\pm) symbol when taking the square root of both sides of an equation, as both positive and negative values can be squared to give a positive result. Therefore, the zeros of our function \(r(x)\) are the square roots of 48, accounting for both the positive and negative possibilities.