The Discriminant and the Number of Roots

Mathematics · Algebra · ages 15-16

Name ______________________ Date ____________

For x squared - 2x - 4 = 0, is the discriminant positive, zero, or negative?
- zero
- positive
- negative
Which of these equations has a repeated root?
- x squared + 4x + 4
- x squared + 4x + 5
- x squared + 4x + 3
The graph of y = 2x squared + 4x + 2 touches the x-axis at exactly one point.
Circle one: True False
What is the discriminant of x squared + 10x + 25?
Answer: ______________
What is the repeated root of x squared - 8x + 16 = 0?
- 4
- 2
- 8
- -4
Which value of k makes x squared + kx + 16 = 0 have a repeated root?
- 16
- 8 and -8
- 8 only
- 4 and -4
Tom works out the discriminant of x squared + 2x + 5 as 4 - 5 = -1 and concludes there are no real roots. What is the slip in his working?
- He subtracted in the wrong order
- His conclusion about the number of roots is wrong
- He squared the wrong coefficient
- He used b squared - c instead of b squared - 4ac
The discriminant of x squared + px + 9 is zero, and p is positive. p is 6.
Circle one: True False
The equation 2x squared + kx + 5 = 0 has no real roots when k squared is below a certain value. What is that value?
Answer: ______________
A quadratic equation with a = 1 has a repeated root at x = -4. What is its constant term c?
- 8
- -16
- 16
- -8

Answer key

For grown-ups. Fold this page away before handing over the rest.

positive · The discriminant is 4 - 4(1)(-4), which is 4 + 16, giving 20, a positive number.
x squared + 4x + 4 · Only x squared + 4x + 4 has a zero discriminant: 16 - 16 is 0, while the others give -4 and 4.
True · The discriminant is 16 - 16 = 0, so there is one repeated root, and a repeated root is a graph that touches the axis once without crossing.
0 · b squared is 100 and 4ac is 100, so the discriminant is 100 - 100, which is 0.
4 · The discriminant is 64 - 64, which is 0, and the repeated root is -b over 2a, which is 8 over 2.
8 and -8 · A repeated root means k squared - 64 is 0, so k squared is 64 and k is 8 or -8.
He used b squared - c instead of b squared - 4ac · The correct working is 4 - 4(1)(5), which is 4 - 20, giving -16. Tom dropped the factor of 4, though his conclusion still happened to be right.
True · The discriminant is p squared - 36, so p squared is 36 and p is 6 or -6. The positive value is 6.
40 · No real roots need k squared - 40 < 0, so k squared must be below 40.
16 · A repeated root at x = -4 means the equation is (x + 4) squared, which expands to x squared + 8x + 16, so c is 16.