Question

Find the zero(s) of the function. (See Example 4.) \(f(x)=x^2-8 x+16\)

Short answer

The only solution or zero of the function \(f(x) = x^2 - 8x + 16\) is \(x = 4\).

Step-by-step solution

Identify the coefficients

In the equation \(f(x) = x^2 - 8x + 16\), identify the coefficients a, b, and c to use in the quadratic formula. In this quadratic equation, \(a = 1\), \(b = -8\), and \(c = 16\).

Use the quadratic formula

Use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), and plug in the values of a, b, and c. This gives us \(x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4*1*16}}{2*1}\).

Simplify the equation

Simplify the equation to find the values of x. So, \(x = \frac{8 \pm \sqrt{64 - 64}}{2} = \frac{8 \pm \sqrt{0}}{2}\). This simplifies further into \(x = \frac{8 \pm 0}{2}= 4\).

Answer

The solution is \(x = 4\). This is the only solution or zero of the function since both roots are the same.

Key concepts

Zeros of a Function

When you hear the term 'zeros of a function,' it simply refers to the values of the variable (often x) when the function equals zero.
These zeros are the x-coordinates where the graph of the function crosses or touches the x-axis. Understanding zeros can provide insight about the function's graph and behavior.

• For a linear function, there can be zero or one zero.
• For a quadratic function, like our given example, there can be either zero, one, or two distinct zeros.

Finding zeros is like solving the equation set to zero, so it's sometimes referred to as solving for the roots of the function.

Quadratic Formula

Sometimes, finding the zeros of a quadratic function isn't straightforward. That's where the quadratic formula comes in handy. This formula provides a surefire method to find the zeros of any quadratic equation in the form of \(ax^2 + bx + c = 0\).
Here's how it works:

• Identify the coefficients: In our equation \(f(x) = x^2 - 8x + 16\), we identify \(a = 1\), \(b = -8\), and \(c = 16\).
• Plug these values into the formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• Simplify to find the x values: This involves operations like squaring \(b\), computing the discriminant \(b^2 - 4ac\), and applying square root and basic arithmetic.

The formula not only gives the zeros but helps determine the nature of the roots, depending on the discriminant value.
If it's positive, you get two distinct roots. If zero, a double root, and negative means no real roots.

Completing the Square

Completing the square is another powerful algebraic technique to find the zeros of a quadratic function, often when factoring or the quadratic formula isn't convenient. This method creates a perfect square trinomial from a quadratic equation.
Here's a general guide on how completing the square works:

• Start with the quadratic equation: In standard form, like \(x^2 - 8x + 16\).
• Move the constant term to the other side, if needed, making it \(x^2 - 8x = -16\).
• Find the number to complete the square: Take half of the b-coefficient (from \(-8\), so \(-4\)), square it (\(16\)), and balance the equation by adding this to both sides.
• Rewrite as a square: The left side becomes \((x - 4)^2\), making it obvious the root is at \(x = 4\).

This method is especially helpful as it not only gives the zeros but helps in rewriting the function's equation in vertex form, thus illuminating more about its graph and vertex.