Name: _______________________
Non Factorable Quadratics – Discriminant
Date: ___________
Algebra I Common Core
Interpreting the Discriminant
Sheet # 17-06
The expression 𝒃𝟐 − 𝟒𝒂𝒄 in the Quadratic Formula is called the discriminant.
Because the discriminant is under the radical symbol, you can use the value of the
discriminant to determine the number of real solutions of a quadratic equation and
the number of x-intercepts of the graph of the related function.
A Quadratic Equation will have…
Two Irrational Solutions
Two Rational Solutions
One Rational Solution [Equal]
No Real Solutions [Imaginary]
and two x-intercepts
and two x-intercepts
and one x-intercept
and no x-intercepts
when the discriminant is
when the discriminant is
when the discriminant is
when the discriminant is
Example: 𝑥 2 + 5𝑥 − 1
Example: 2𝑥 2 + 7𝑥 − 4
Example: 𝑥 2 + 4𝑥 + 4
Example: 𝑥 2 − 6𝑥 + 10
𝑥=
−(5) ± √(5)2 − 4(1)(−1)
2(1)
𝑥=
−(7) ± √(7)2 − 4(2)(−4)
2(2)
𝑥=
−(4) ± √(4)2 − 4(1)(4)
2(1)
𝑥=
−(−6) ± √(−6)2 − 4(1)(10)
2(1)
𝑥=
−5 ± √29
2
𝑥=
−7 ± √81
4
𝑥=
−4 ± √0
2
𝑥=
6 ± √−4
2
Discriminant: ___________
𝑥=
−5+√29
2
or 𝑥 =
−5−√29
2
x = _________ or x = _________
Discriminant: ___________
𝑥=
−7+9
4
or
𝑥=
−7−9
4
x = _________ or x = _________
Discriminant: ___________
𝑥=
−4+0
2
or
𝑥=
−4−0
2
x = _________ or x = _________
Discriminant: ___________
𝑥=
6+√−4
2
or 𝑥 =
6−√−4
2
x = _________ or x = _________
1
Match each discriminant in the chart with the appropriate description of the nature of the roots and the possible graph.
Discriminant
b 2  4ac
Nature of the Roots
Possible Graph
Nature of the Roots
I.
Two Real and Rational Roots
II.
Two Real and Irrational Roots
III.
One Real and Rational Root
IV.
No Real Solutions or Imaginary Solutions
0
25
-25
54
Graph A
Graph C
Graph D
Use the graph to justify:
How many real solutions does the
equation 4𝑥 2 + 2𝑥 + 3 have?
Justify your answer.
Use the graph to justify:
How many real solutions does the
equation 4𝑥 2 + 4𝑥 + 1 have?
Justify your answer.
Use the graph to justify:
Use the discriminant to justify:
Use the discriminant to justify:
Use the discriminant to justify:
2
How many real solutions does the
equation 𝑥 2 − 5𝑥 − 3 have?
Justify your answer.
Graph B
3
4