Start by identifying the values of a, b, and c.
Be sure that all of the terms of are on the same side and in the correct order for the standard form of a quadratic function.
See solution.
Practice makes perfect
To determine the number of real zeros of f(x)=x^2-8x+16, we will solve the equation f(x)=0 by factoring.
f(x)=0 ⇔ x^2-8x+16=0
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.
x^2-8x+16=0 ⇕ 1x^2+( - 8)x+ 16=0
We have a quadratic equation with a= 1, b= - 8, and c= 16. To factor the left-hand side, we need to find a factor pair of 1 * 16=16 whose sum is - 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
^(1* 16) 16
1+16 17
- 1 and - 16
^(- 1* (- 16)) 16
- 1+(- 16) - 17
2 and 8
^(2* 8) 16
2+8 10
- 2 and - 8
^(- 2* (- 8)) 16
- 2+(- 8) - 10
4 and 4
^(4* 4) 16
4+4 8
- 4 and - 4
^(- 4* (- 4)) 16
- 4+(- 4) - 8
The integers whose product is 16 and whose sum is - 8 are - 4 and - 4. With this information, we can rewrite the linear factor on the left-hand side of the equation, and factor by grouping.
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.
ax^2+ bx+ c=0 ⇕ x=- b± sqrt(b^2-4 a c)/2 a
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= - 8, and c= 16. Let's find the discriminant.
