The zeros of the function f(x)=x^2-3x+2 are equal to the roots of its associated equation x^2-3x+2=0.
Zeros: x=1 and x=2 How can you verify? See solution.
Practice makes perfect
Recall that the zeros of a quadratic function are equal to the roots of it associated equation. Let's take a look at the function given.
Quadratic Function 0.55cm Associated Equation
f(x)=x^2-3x+2 0.75cm x^2-3x+2 = 0
Since the associated equation is in standard form, we can factor it and solve it by using the Zero Product Property. Note that the left-hand side is a trinomial of the form x^2+bx+c.
x^2+ bx+ c
x^2+( -3)x+ 2
Therefore, we can factor it by finding two numbers that add up to b= -3 and whose product is c = 2. Since -1 +(- 2)=- 3 and (- 1)(- 2) = 2, we can use them to factor the trinomial. By doing this we obtain an equivalent quadratic equation.
x^2 -3 x+2 =0
⇕
(x-1)(x-2)=0
Now we can use the Zero Product Property.
Zero Product Property
For any two real number a and b, if ab=0 then a=0 or b=0.
According to this property, if (x-1)(x-2)=0, then x-1=0 or x-2=0. Therefore, the solutions to these equations are also solutions to the original quadratic equation. We can solve them with one step.
x-1=0 ⇔ x=1 [0.5em]
x-2=0 ⇔ x=2
Thus, the solutions to x^2-3x+2 =0 are x=1 and x=2. As we already mentioned, these are also the zeros of the function. We can verify this by graphing. To do this, recall that the zeros of a function equal the x-intercepts its graph.
As we can see, the graph intersects the x-axis at ( 1,0) and ( 2,0). Therefore, the x-intercepts, as well as the zeros of the function, are x= 1 and x= 2. This confirms our previous results.
