find the zeros of f(x) = x^2-8x+16

We are asked to find the zeros of the following quadratic function, f(x) = x² - 8x + 16, where a = 1, b = -8, and c = 16. We will be using the following methods for finding the zeros of the given quadratic function: using the quadratic equation and factoring the expression.

Definitions:

The given quadratic function is written in its standard form, f(x) = ax² + bx + c where a ≠ 0.
The zeros (also known as roots ) refers to the solutions of a quadratic function. These are also the x-intercepts of the graph, which are the points where the parabola crosses the x-axis.

The discriminant (under the radical of the quadratic formula), "b² - 4ac" determines the number of solutions:

b² - 4ac > 0 ⇒ The function has two real-number solutions.
b² - 4ac = 0 ⇒ The function has one real-number solution.
b² - 4ac < 0 ⇒ The function has no real-number solution.

Solution:

Method 1: Using the Quadratic Equation

We can use the quadratic equation to solve for the roots of the given function: ⇒ [tex]\displaystyle\mathsf{x\:=\frac{-b\pm\sqrt{\:b^2-4ac}}{2a}}[/tex]

Step 1: Substitute the values for a, b, and c into the quadratic formula:

f(x) = x² - 8x + 16 ⇒ where a = 1, b = -8, and c = 16.

⇒ [tex]\displaystyle\mathsf{x\:=\frac{-(-8)\pm\sqrt{\:(-8)^2-4(1)(16)}}{2(1)}}[/tex]

Step 2: Use the PEMDAS order of operations and simplify:

⇒ [tex]\displaystyle\mathsf{x\:=\frac{8\pm\sqrt{\:64-64}}{2}\quad\longrightarrow\quad=\frac{8\pm\sqrt{\:0}}{2}\quad=\quad\frac{8}{2}\quad=\quad4}[/tex]

Therefore, the root or solution to the given quadratic function is x = 4.

Method 2: Factoring the Expression

In this method, we must find the factors of the given quadratic function. Then, we will use those factors and apply the Zero-Product Property Rule to solve for the zeros or roots of the function.

Step 1: Set the function equal to zero.

f(x) = x² - 8x + 16

⇒ 0 = x² - 8x + 16

Step 2: Find factors whose product results in a · c = 16, and the sum of b = - 8.

product of a · c = 16 ⇒ (-4) · (-4) = 16
sum of b = - 8 ⇒ (-4) + (-4) = - 8

Hence, our possible factors are: 0 = (x - 4)(x - 4) or 0 = (x - 4)²

Step 3: Apply the Zero-Product Property Rule, which states that for any real numbers m and n, if mn = 0, then m = 0 or n = 0.

(x - 4)² = 0

⇒ [tex]\displaystyle\mathsf{\sqrt{(x-4)^2}\:=\: \sqrt{0}}[/tex]

⇒ x - 4 = 0

⇒ x - 4 + 4 = 0 + 4

⇒ x = 4

Final Answer:

Based from both methods used in this post, we can infer that the root or solution to the given quadratic function, f(x) = x² - 8x + 16 is x = 4. This means that The attached screenshot shows that the graph crosses the x-axis only once, at (4, 0), which also happens to be its vertex (minimum point).

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Keywords:

Quadratic functions

Quadratic equations

Parabola

Roots

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