Question

Find the zero(s) of the function. (See Example 4.) \(g(x)=x^2+6 x+8\)

Short answer

The zeros of the function \(g(x)=x^2+6 x+8\) are \(x=-2\) and \(x=-4\).

Step-by-step solution

Identify the problem

We are tasked to find the zero(s) of the function \(g(x)=x^2+6 x+8\). That means we must find the value(s) of \(x\) which make the entire function equal to zero.

Set the function equal to zero

To find the zeros of the function, we set the function equal to zero: \(x^2+6 x+8=0\). This equation can be factored by looking for a pair of numbers that multiply to 8 (the third coefficient) and add up to 6 (the second coefficient).

Factor the quadratic equation

We identify the pair of numbers as 2 and 4, because 2 and 4 multiply to yield 8, and also add up to 6. This lets us factor our equation as follows: \((x+2)(x+4)=0\).

Solve for x

Apply the zero-product property, which states that if a product of factors equals zero, at least one of the factors must be zero. Setting each factor equal to zero, we get \(x+2=0\) or \(x+4=0\). Solving these equations yield \(x=-2\) or \(x=-4\), respectively.

Key concepts

Zero-Product Property

Understanding the zero-product property is crucial when dealing with quadratic functions. It states that if a product of two or more factors is zero, then at least one of the factors must equal zero. For example, if we have \(a \times b = 0\), then \(a=0\) or \(b=0\), or both might be true.
When applied to solving quadratic equations, this property becomes a powerful tool. After factoring the quadratic equation into two binomials, as done in the exercise \( (x+2)(x+4)=0 \), we use the zero-product property to set each factor to zero, leading to the solutions of the equation. \( x+2=0 \) gives us \( x=-2 \), while from \( x+4=0 \) we find \( x=-4 \). These solutions are the 'zeros' or 'roots' of the quadratic function, corresponding to the values where the graph of the function intersects the x-axis.

Factoring Quadratic Equations

Factoring is a method of breaking down the quadratic equation \( ax^2+bx+c=0 \) into a product of two binomials. In our exercise, the equation \( x^2+6x+8=0 \) is factored into \( (x+2)(x+4)=0 \). This is possible because we can find two numbers that both add up to the coefficient 'b' and multiply together to give the constant term 'c'.
In a simple step-by-step approach:

• Determine two numbers that multiply to 'c' (the constant term) and add to 'b' (the coefficient of x).
• Write the original quadratic as a product of binomials using these numbers.
• The factored form makes it easier to apply the zero-product property and find the zeros of the function.

It's also worth noting, not all quadratic equations can be easily factored, and other methods may be required, such as completing the square or using the quadratic formula.

Solving Quadratic Equations

Once a quadratic equation is factored, the next step is to solve it to find the values of x that make the equation true. In our example, after factoring the equation \( g(x)\), we solve the resulting binomials \( x+2=0 \) and \( x+4=0 \).
Completing these simple steps provides the zeros of the quadratic function:

• If \( x+2=0 \), subtract 2 from both sides to get \( x = -2 \).
• If \( x+4=0 \), subtract 4 from both sides to get \( x = -4 \).

These are the values where the function \( g(x) \) will cross the x-axis when graphed. Always check your solutions by substituting them back into the original equation to ensure that they satisfy the equation. In cases where factoring is not possible, other methods such as the quadratic formula \( x = \frac{-b \bin \sigmaqrt{b^2 - 4ac}}{2a} \) are used to find the zeros of the function.