MPM2D
Ms. Kueh
Number of Roots of a Quadratic Equation
Investigation
Use the quadratic formula to find the zeros of each parabola:
a) 𝑦 = 𝑥 2 + 2𝑥 − 9
b) 𝑦 = 2𝑥 2 − 12𝑥 + 18
c) 𝑦 = 3𝑥 2 + 4
1. Graph the following parabolas on the grid below using any method from the last unit.
a) 𝑦 = 𝑥 2 + 2𝑥 − 9
b) 𝑦 = 2𝑥 2 − 12𝑥 + 18
c) 𝑦 = 3𝑥 2 + 4
Definition: __________________ is called the __________________________. It tells us
how many roots a quadratic equation will have.
If the _______________ is positive, there will be __________________.
If the _______________ is zero, there will be _____________________.
If the _______________ is negative, there will be _________________.
Example 1
Determine the number of roots each equation
2
a) −5𝑥 + 8𝑥 − 10 = 0
b) 7𝑥 2 − 6 = 0
Example 2
Find the complex roots of:
Homework: Worksheet
𝑦 = 𝑥 2 − 4𝑥 + 7
Discriminant and Quadratic Formula Worksheet
1. Determine how many real roots the equation has.
a. 𝑥 2 + 9 = 0
b. 𝑥 2 + 5𝑥 − 8 = 0
c. 2𝑥 2 − 12 = 0
d. 3𝑥 2 + 2𝑥 + 6 = 0
e. 9𝑥 2 + 12𝑥 = −4
f. 𝑥 2 = −8𝑥 + 3
g. (𝑥 − 4)2 = 0
h. −3(𝑥 + 2)2 + 10 = 0
i. (𝑥 − 2)2 − 5 = 0
2. For g, h, and i there are 2 methods of finding the number of roots. Find the
number of roots an alternate way. Which way required less work?
3. Find the roots of the equation by factoring.
a. 𝑥 2 − 9 = 0
b. 𝑥 2 + 𝑥 − 6 = 0
c. 2𝑥 2 − 8𝑥 = 0
d. 2𝑥 2 + 𝑥 − 15 = 0
e. 3𝑥 2 − 75 = 0
4. Why is factoring preferable to using quadratic formula in question 3?
5. Find the roots of the equation, if possible. Use the most appropriate method.
a. 𝑥 2 − 8𝑥 = −16
b. 2𝑥 2 + 3𝑥 − 20 = 0
c. (𝑥 − 5)2 = 16
Hint: move everything to the other side to isolate 𝑥
d. 𝑥 2 + 10 = 0
e. −2(𝑥 + 1)2 + 10 = 0
f. 𝑥 2 = 90 − 6𝑥
g. −5𝑥 2 + 15𝑥 = 11
h. 3.2𝑤 2 + 28.9𝑤 − 8.4 = 0
6. Determine the number and type of roots (real and distinct, real and equal, or
complex)
a.
b.
𝑥2
2
+4𝑥 + 4 = 0
𝑥−1
2
−𝑥 2 − 3 = 0
c. (𝑥 + 1)(𝑥 − 2) = 4
d. 4(𝑥 2 − 5𝑥 + 5) = −5
TIPS Practice:
7. Determine the value(s) of 𝑘 that will give the indicated solution.
a. 𝑥 2 − 4𝑥 + 𝑘 = 0
; equal roots
b. 𝑥 2 + 3𝑥 − 2𝑘 = 0
; complex roots
c. 𝑥 2 + 𝑘𝑥 + 16 = 0
; real distinct roots
8. Find the value of 𝑘 such that the quadratic equation 𝑘𝑥 2 + (𝑘 + 8)𝑥 + 9 = 0
has equal roots.
Answers:
1. a) none
b) 2
c) 2
d) none
e) 1
f) 2
2. Sketch the parabola, since it is in vertex form. If it is already in vertex form, it is
usually less effort to just do a quick sketch.
3. a) 𝑥 = −3,3
b) 𝑥 = −3,2
c) 𝑥 = 0,4
5
d) 𝑥 = 2 , −3 e) 𝑥 = −5, 5
4. It is much faster, and less likely that you will make a mistake.
5. a) 4
e) 1.24, -3.24
b) 2.5, -4
c) 9, 1
f) 6.95, -12.95
g) 1.72, 1.28 h) 0.28, -9.31
6. a) real and distinct
b) complex
7. a) 4
b)𝑘 < − 8
8. 4 or 16
9
c) real and distinct
c) 𝑘 > 8 or 𝑘 < −8
d) no real roots
d) real and equal