Question

Find the zeros of \(f(x)=x^{2}+6 x-8\)

Short answer

The zeros are \(x = -8\) and \(x = 2\).

Step-by-step solution

Identify the quadratic equation

The given function is a quadratic equation in the standard form: \[f(x) = x^2 + 6x - 8\]We need to find the values of \(x\) for which the function equals zero, i.e., \(f(x) = 0\).

Set the equation to zero

Set the quadratic function equal to zero:\[x^2 + 6x - 8 = 0\]

Factor the quadratic equation

To solve the equation by factoring, find two numbers that multiply to -8 and add to 6. The numbers 8 and -2 work because:\[8 \times (-2) = -8\]\[8 + (-2) = 6\]So, we can factor the quadratic as:\[x^2 + 6x - 8 = (x + 8)(x - 2)\]

Solve for the zeros

Set each factor equal to zero and solve for \(x\):\[x + 8 = 0 \rightarrow x = -8\]\[x - 2 = 0 \rightarrow x = 2\]

Key concepts

Factoring Quadratics

Factoring quadratics means expressing the quadratic equation in the form of a product of its linear factors. Given the quadratic equation \(x^2 + 6x - 8\), we aim to rewrite it as a product of two binomials.

To do this, we look for two numbers that multiply to the constant term (in this case, -8) and add up to the coefficient of the linear term (here, 6).

In the equation \(x^2 + 6x - 8\), the two numbers that work are 8 and -2 because 8 \( \times \) -2 = -8 and 8 + (-2) = 6. This gives us the factors (x + 8) and (x - 2).

Therefore, we can write: \[x^2 + 6x - 8 = (x + 8)(x - 2)\]

Factoring is crucial because it simplifies the process of finding solutions to the quadratic equation.

Finding Zeros

Finding the zeros of a quadratic function means determining the values of \x\ where the function equals zero. For the given function \(f(x)=x^2 + 6x - 8\), we are interested in the values of \x\ that make \f(x) = 0\.

After factoring the quadratic equation \(x^2 + 6x - 8\) into \((x + 8)(x - 2)\), we set each factor equal to zero. This step is based on the Zero Product Property, which states that if the product of two numbers is zero, at least one of the numbers must be zero.

Setting each factor to zero will look like this:

• \x + 8 = 0 \rightarrow x = -8\
• x - 2 = 0 \rightarrow x = 2 \

This results in the solutions \x = -8\ and \x = 2\, which are the zeros of the function. This means that the function \(f(x)\) crosses the x-axis at these points.

Solving Quadratic Equations

Solving a quadratic equation involves finding the values of \x\ that make the equation true. For the quadratic equation \(x^2 + 6x - 8 = 0 \), we have already factored it to get \((x + 8)(x - 2) = 0\).

The next steps involved using the factors to find the solutions by setting each factor equal to zero:
\[x + 8 = 0 \rightarrow x = -8\]
\[x - 2 = 0 \rightarrow x = 2\]

Thus, the solutions to the quadratic equation \(x^2 + 6x - 8 = 0\) are \x = -8\ and \x = 2\. These solutions are where the graph of the quadratic function intersects the x-axis.

It's important to recognize that solving quadratic equations can be approached in different ways, such as factoring, using the quadratic formula, or completing the square. In this problem, factoring was the best method because the quadratic easily factored into two linear binomials.