Find The Quadratic Discriminant - Worksheet - Free Printable

To solve the problems, we need to use the formula for the discriminant of a quadratic equation. The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

The discriminant (\(\Delta\)) is given by:

\[
\Delta = b^2 - 4ac
\]

Problem 1: \( k^2 + 4k + 3 = 0 \)

Here, \(a = 1\), \(b = 4\), and \(c = 3\).

\[
\Delta = b^2 - 4ac = 4^2 - 4(1)(3) = 16 - 12 = 4
\]

Problem 2: \( n^2 + 11n + 10 = 0 \)

Here, \(a = 1\), \(b = 11\), and \(c = 10\).

\[
\Delta = b^2 - 4ac = 11^2 - 4(1)(10) = 121 - 40 = 81
\]

Problem 3: \( x^2 + 14x + 40 = 0 \)

Here, \(a = 1\), \(b = 14\), and \(c = 40\).

\[
\Delta = b^2 - 4ac = 14^2 - 4(1)(40) = 196 - 160 = 36
\]

Problem 4: \( n^2 - 11n + 24 = 0 \)

Here, \(a = 1\), \(b = -11\), and \(c = 24\).

\[
\Delta = b^2 - 4ac = (-11)^2 - 4(1)(24) = 121 - 96 = 25
\]

Problem 5: \( y^2 + 8y + 29 = 0 \)

Here, \(a = 1\), \(b = 8\), and \(c = 29\).

\[
\Delta = b^2 - 4ac = 8^2 - 4(1)(29) = 64 - 116 = -52
\]

Problem 6: \( x^2 + 20x + 107 = 0 \)

Here, \(a = 1\), \(b = 20\), and \(c = 107\).

\[
\Delta = b^2 - 4ac = 20^2 - 4(1)(107) = 400 - 428 = -28
\]

Problem 7: \( a^2 + 14a + 52 = 0 \)

Here, \(a = 1\), \(b = 14\), and \(c = 52\).

\[
\Delta = b^2 - 4ac = 14^2 - 4(1)(52) = 196 - 208 = -12
\]

Problem 8: Determine the number of real solutions for \( 5n^2 + 11n + 16 = 0 \)

Here, \(a = 5\), \(b = 11\), and \(c = 16\).

\[
\Delta = b^2 - 4ac = 11^2 - 4(5)(16) = 121 - 320 = -199
\]

Since the discriminant is negative (\(\Delta < 0\)), there are no real solutions.

Final Answers

1. \(\boxed{4}\)
2. \(\boxed{81}\)
3. \(\boxed{36}\)
4. \(\boxed{25}\)
5. \(\boxed{-52}\)
6. \(\boxed{-28}\)
7. \(\boxed{-12}\)
8. \(\boxed{0}\) (No real solutions)