Problem 51 - Precalculus : Building Concepts and Connections

Question

Compute the zeros of the quadratic function. $$h(x)=-3 x^{2}-10$$

Step-by-Step Solution

Answer

The quadratic function \(h(x) = -3x^2 -10\) has no real roots.

Step 1: Set the function to zero

In order to find the zeros of the function, we need to set the function equal to zero. So, we have \(0 = -3x^2 -10\).

Step 2: Solve for \(x^2\)

Move -10 over to the other side to isolate \(x^2\). It will be \(-3x^2 = 10\), then divide by -3 both sides to get \(x^2 = -\frac{10}{3}\).

Step 3: Square root both sides

Since \(x^2 = -\frac{10}{3}\), we can now find the value of x by applying square root on both sides. However, the square root of a negative number isn't real, it's imaginary. Therefore, this quadratic function has no real roots.

Key Concepts

Zeros of a Function

The zeros of a function, sometimes called the roots of the function, are the values of the variable that make the function equal to zero. For a quadratic function like \(h(x)=-3x^2-10\), the zeros are the \(x\)-values where \(h(x) = 0\). To find these, set the entire equation equal to zero, resulting in the equation \(0 = -3x^2 -10\). This means we are looking for values of \(x\) such that the expression on the right equals zero.

Finding zeros is essential because they represent the points at which a graph intersects the x-axis. For real-world applications, zeros can show when a particular situation or condition modeled by the function is in balance or at equilibrium. Understanding where these points are helps anticipate and plan for shifts in the modeled scenario.

• Quadratic functions typically have two zeros or no real zeros.
• If the zeros are real, they will appear where the graph of the function crosses the x-axis.

Imaginary Numbers

Imaginary numbers come into play when we deal with the square roots of negative numbers. In our given quadratic \(h(x)=-3x^2-10\), solving for the zeros leads us to \(x^2 = -\frac{10}{3}\). Here, a crucial point is that there is no real number that can be squared to yield a negative, which introduces the concept of imaginary numbers.

The basic imaginary unit is denoted as \(i\), where \(i = \sqrt{-1}\). For our purposes, this means that taking the square root of both sides of \(x^2 = -\frac{10}{3}\) involves introducing \(i\).

• For a negative number inside a square root, we rewrite the square root of the negative as the square root of the positive number multiplied by \(i\). For instance, \(\sqrt{-\frac{10}{3}} = \sqrt{\frac{10}{3}}\cdot i\).
• Imaginary numbers are crucial in extending the real number system and are often used in engineering and physics.

Solving Quadratic Equations

Solving a quadratic equation involves finding the values of \(x\) that satisfy the equation. The given exercise provides the quadratic equation \(0 = -3x^2 - 10\). Here's how to approach solving such equations.

First, you isolate \(x^2\) by moving constants over to the other side of the equation. This typically involves adding or subtracting terms on both sides. For our inventory, we moved \(-10\) from the right, leaving us with \(-3x^2 = 10\). The next step is to divide all terms by the coefficient of \(x^2\) (which is \(-3\) in this case), resulting in \(x^2 = -\frac{10}{3}\).

Finally, you take the square root of both sides to solve for \(x\). However, because we end up with a negative number under the square root, we find our solutions in terms of imaginary numbers: \(x = \pm \sqrt{-\frac{10}{3}}\) which are \(x = \pm \sqrt{\frac{10}{3}} \cdot i\).

• Quadratic equations can have two real solutions, one real solution, or two complex conjugate solutions.
• Understanding whether a quadratic will result in real or imaginary numbers is part of the qualities that differentiate equations.