Exercise 52 Page 588

To determine the number of real zeros of $f(x)=x^2-8x+16,$ we will solve the equation $f(x)=0$ by factoring. Then we can compare this method to finding the number of real zeros by using the discriminant.

Factoring

To solve the equation by factoring, we will start by identifying the values of $a,$ $b,$ and $c.$ We have a quadratic equation with $a=1,$ $b=\text{-}8,$ and $c=16.$ To factor the left-hand side, we need to find a factor pair of $1\times 16=16$ whose sum is $\text{-}8.$ Since $16$ is a positive number, we will only consider factors with the same sign — both positive or both negative — so that their product is positive.

Factor Pair	Product of Factors	Sum of Factors
$1$ and $16$	$\stackrel{{}^{1\times 16}}{16}$	$\stackrel{1+16}{17}$
$\text{-}1$ and $\text{-}16$	$\stackrel{{}^{\text{-}1\times (\text{-}16)}}{16}$	$\stackrel{\text{-}1+(\text{-}16)}{\text{-}17}$
$2$ and $8$	$\stackrel{{}^{2\times 8}}{16}$	$\stackrel{2+8}{10}$
$\text{-}2$ and $\text{-}8$	$\stackrel{{}^{\text{-}2\times (\text{-}8)}}{16}$	$\stackrel{\text{-}2+(\text{-}8)}{\text{-}10}$
$4$ and $4$	$\stackrel{{}^{4\times 4}}{16}$	$\stackrel{4+4}{8}$
$\text{-}4$ and $\text{-}4$	$\stackrel{{}^{\text{-}4\times (\text{-}4)}}{16}$	$\stackrel{\text{-}4+(\text{-}4)}{\text{-}8}$

The integers whose product is $16$ and whose sum is $\text{-}8$ are $\text{-}4$ and $\text{-}4.$ With this information, we can rewrite the linear factor on the left-hand side of the equation, and factor by grouping. Now we can solve for $x.$ We found that the only unique solution to the given equation is $x=4.$ Therefore, the number of real zeros of $f(x)=x^2-8x+16$ is one.

Discriminant

The discriminant is the expression under the radical sign in the Quadratic Formula, $b^2-4ac.$ Let's analyze the Quadratic Formula. We know that the discriminant can be used to determine the number of real solutions of a quadratic equation by looking at the sign of the result.

Discriminant	Number of Real Solutions
Negative	None
Zero	One
Positive	Two

In our case, $a=1,$ $b=\text{-}8,$ and $c=16.$ Let's find the discriminant. Since the discriminant is $0,$ the number of real zeros of $f(x)=x^2-8x+16$ is one.

Comparison

Let's look at the similarities and differences between the methods.

• Similarities.
  • Both methods are algebraic.
  • We transform algebraic expressions, substitute, and perform various types of transformations.
• Differences.
  • The first method is much longer and requires more mathematical calculations.
  • The second method only requires evaluating the discriminant and checking its sign.