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Chapter 6: Problem 103

Find the zeros of each function. $$ f(x)=x^{2}-16 $$

Short Answer

Expert verified
The zeros are x = 4 and x = -4.

Step by step solution

01

Identify the Function

Examine the given function: \[ f(x) = x^2 - 16 \]This is a quadratic function.
02

Rewrite the Function

Set the function equal to zero to find the zeros: \[ x^2 - 16 = 0 \]
03

Factor the Quadratic Equation

Recognize that the quadratic equation is a difference of squares and can be factored: \[ x^2 - 16 = (x - 4)(x + 4) \]
04

Solve for Zeros

Set each factor equal to zero and solve for x: \[ x - 4 = 0 \quad or \quad x + 4 = 0 \]\[ x = 4 \quad or \quad x = -4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations
Quadratic equations are expressions that can be written in the standard form of \[ax^2 + bx + c = 0\]. In our example, the quadratic equation is \[x^2 - 16 = 0\]. The 'a' term is 1, the 'b' term is 0, and the 'c' term is -16. Quadratic equations graph as parabolas, which can either open upwards or downwards.

To find the zeros of a quadratic function like \[f(x) = x^2 - 16\], we look for the values of 'x' that make the function equal to zero. These points, called the roots or solutions, are where the graph intersects the x-axis.
Factoring
Factoring is the process of rewriting an equation as a product of its simpler components or factors. For quadratic equations, we often look for two binomials that multiply together to give us the original quadratic equation.

In our example, \[x^2 - 16\] can be factored by recognizing it as a difference of squares. The factorization step rewrites the quadratic equation as \[(x - 4)(x + 4)\]. Factoring is essential because it simplifies solving the equation by breaking it down into simpler parts.
Difference of Squares
The difference of squares is a special type of factoring where we have an equation in the form \[a^2 - b^2\]. These can be factored into \[(a - b)(a + b)\]. In our example, we have \[x^2 - 16\]. Notice that 16 is \[(4^2)\], so the equation fits the difference of squares pattern: \[x^2 - 4^2 = (x - 4)(x + 4)\].

Using this method, we can easily factor quadratic equations of this specific type, making the solutions easier to find.
Solving Equations
Once we've factored the quadratic equation, we set each factor equal to zero to find the solutions. This step is based on the zero product property, which states if \[a \cdot b = 0\], then either \[a = 0\] or \[b = 0\].

For the equation \[(x - 4)(x + 4) = 0\], we set each factor to zero and solve: \[x - 4 = 0\] and \[x + 4 = 0\]. Solving these gives us \[x = 4\] and \[x = -4\]. Therefore, the zeros of the function \[f(x) = x^2 - 16\] are 4 and -4.

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Most popular questions from this chapter

Factor completely. Assume that variables in exponents represent positive integers. $$ 18 x^{3}-\frac{8}{25} x $$

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Factor completely. Assume that variables in exponents represent positive integers. $$ 9 x^{2 n}-6 x^{n}+1 $$

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