U4L6 Finding the Zeros - Quadratic Formula
Quadratic Functions Day 6
Finding the Zeros (x-intercepts) of a Quadratic Function
On the graph of a quadratic function, the x-intercept(s) are called the zeros (because that is where the
y value is zero). It is possible for a quadratic (parabola) to have 0, 1, or 2 x-intercepts, so keep the
shape of the quadratic in mind when you are seeking the zeros.
1. On the three grids below, sketch parabolas that have 0, 1, and 2 x-intercepts as labeled.
2 x-intercepts
1 x-intercept
no x-intercepts
2. Up until now we have found the zeros by using Diamond Problems and factoring. But sometimes
the numbers aren’t so nice and we can’t easily factor. When we are unable to factor a quadratic in
standard form, there is a handy formula that mathematicians long before us derived: the quadratic
formula. If the quadratic is of the form y = ax2 + bx + c, we can use the quadratic formula:
x-coordinate(s) of the zeros =
− b ± b 2 − 4ac
2a
For each quadratic function, find the x-intercepts by using the quadratic formula.
a. 𝑓 𝑥 = 2𝑥 ! − 5𝑥 + 2
b. 𝑓 𝑥 = −𝑥 ! − 6𝑥 − 9
c. 𝑓 𝑥 = !!𝑥 ! − 3
d. 𝑓 𝑥 = 𝑥 ! − 9𝑥
U4L6 Finding the Zeros - Quadratic Formula
3. What happens where there are no x-intercepts? Let’s look at the algebra behind it.
a. Vertex Form. For the function 𝑓 𝑥 = 2(𝑥 − 3)! + 1,
sketch a graph using the idea of transformations. Plot
the vertex and a couple of other key points along with
the line of symmetry.
Now let’s try to solve for the x-intercepts
algebraically. We know there are no x-intercepts, but
what happens when we try to solve for them anyway?
This method will help when we don’t have the graph.
Use the quadratic formula to try to find the zeros,
even though we know there aren’t any.
What happened? How does the algebra tell us that there are no x-intercepts? Explain in a
sentence or two.
b. Standard Form & the Discriminant. For a quadratic equation in standard form, we usually try to
factor first to find the x-intercepts.
The following quadratic equation does not have any x-intercepts. First try to factor it by
thinking about a diamond problem, then find the x-intercepts using the quadratic formula.
𝑔 𝑥 = 𝑥 ! − 2𝑥 + 4
What happened, and how does that tell you that this function has no x-intercepts? Explain in a
sentence or two.
U4L6 Finding the Zeros - Quadratic Formula
Find the vertex, axis of symmetry, and y-intercept of the
function 𝑔 𝑥 = 𝑥 ! − 2𝑥 + 4, then sketch it on the graph at
right. Check your work by graphing it on your TI-84
The discriminant, 𝑏 ! − 4𝑎𝑐 , is the part of the quadratic formula tells us whether the function
will have 0, 1, or 2 x-intercepts. Fill in the following key statements below:
When the discriminant is negative (𝑏 ! − 4𝑎𝑐 < 0), the function will have ______ x-intercepts
When the discriminant is zero (𝑏 ! − 4𝑎𝑐 = 0), the function will have ______ x-intercepts
When the discriminant is positive (𝑏 ! − 4𝑎𝑐 > 0), the function will have ______ x-intercepts
4. Group Practice Exercises.
a. For each of the following quadratic functions, state how many x-intercepts (zeros) its graph will have. Note:
You don’t actually need to find the x-intercepts (zeros)!!
!
i) 𝑓 𝑥 = − ! 𝑥 ! + 2𝑥 − 2
ii) 𝑓 𝑥 = 2𝑥 ! − 8𝑥 + 9
iii) 𝑓 𝑥 = −4𝑥 ! − 8𝑥 − 3
b. Find the values of m for which 2x2 – 3x + m = 0 has:
i) only one x-intercept
ii) two x-intercepts
iii) no x-intercepts
U4L6 Finding the Zeros - Quadratic Formula
c. Use the quadratic formula to solve (i.e. find the x-intercepts) ONLY IF you cannot factor! (Remember that to find
the x-intercepts, one side of the equation needs to be equal to zero!
i)𝑥 ! + 6𝑥 + 9 = 0 ii)𝑥 ! + 4𝑥 + 3 = 0iii)𝑥 ! = 10𝑥 − 17