Prepared by Muhammad Tayyab, Subject Specialist Mathematics, Govt Christian High School Daska
📌 Based on National Curriculum 2023 / PECTAA 2026 Syllabus
📖 What's Inside: This exercise covers examination of nature of roots using discriminant, finding parameter values for real/equal/imaginary roots. Perfect for Punjab Boards exam preparation.
📚 Related Resources – Unit 2: Quadratic Equations & Inequalities
1 Examine the nature of roots
(i) \(3x^{2} - 9x - 2 = 0\)
Here, \(a = 3, b = -9, c = -2\)
\[ \text{Disc.} = b^{2} - 4ac = (-9)^{2} - 4(3)(-2) = 81 + 24 = 105 > 0 \]As Disc. \(>0\) and not a perfect square → roots are irrational and unequal.
(ii) \(x^{2} + 6x + 9 = 0\)
\(a=1, b=6, c=9\)
\[ \text{Disc.} = 36 - 4(1)(9) = 36 - 36 = 0 \]Disc. \(=0\) → roots are rational (real) and equal.
(iii) \(2x^{2} + 4x + 5 = 0\)
\(a=2, b=4, c=5\)
\[ \text{Disc.} = 16 - 4(2)(5) = 16 - 40 = -24 < 0 \]Disc. \(<0\) → roots are imaginary and unequal.
(iv) \(7x^{2} - 6x - 1 = 0\)
\(a=7, b=-6, c=-1\)
\[ \text{Disc.} = 36 - 4(7)(-1) = 36 + 28 = 64 > 0 \]Disc. \(>0\) and perfect square → roots are rational and unequal.
(v) \(5x^{2} - 2x + 10 = 0\)
\(a=5, b=-2, c=10\)
\[ \text{Disc.} = 4 - 4(5)(10) = 4 - 200 = -194 < 0 \]Disc. \(<0\) → roots are imaginary and unequal.
(vi) \(x^{2} - 8x + 16 = 0\)
\(a=1, b=-8, c=16\)
\[ \text{Disc.} = 64 - 4(1)(16) = 64 - 64 = 0 \]Disc. \(=0\) → roots are rational (real) and equal.
2 Find values of \(t\) for real and unequal roots
\(3x^{2} + x + 9t = 0\)
\(a=3, b=1, c=9t\)
\[ \text{Disc.} = 1 - 4(3)(9t) = 1 - 108t \]For real and unequal roots: Disc. \(>0\)
\[ 1 - 108t > 0 \quad\Rightarrow\quad 1 > 108t \quad\Rightarrow\quad t < \frac{1}{108} \]\(\boxed{t < \frac{1}{108}}\)
3 Find \(p\) for equal roots
\(16x^{2} + 7px + 49 = 0\)
\(a=16, b=7p, c=49\)
\[ \text{Disc.} = (7p)^{2} - 4(16)(49) = 49p^{2} - 3136 \]For equal roots: Disc. \(=0\)
\[ 49p^{2} - 3136 = 0 \quad\Rightarrow\quad 49p^{2} = 3136 \quad\Rightarrow\quad p^{2} = 64 \] \[ p = \pm 8 \]\(\boxed{p = 8 \text{ or } p = -8}\)
4 Possible values of \(q\) for real and unequal roots
\(4u^{2} + 8u + q = 0\)
\(a=4, b=8, c=q\)
\[ \text{Disc.} = 64 - 4(4)(q) = 64 - 16q \]For real and unequal: Disc. \(>0\)
\[ 64 - 16q > 0 \quad\Rightarrow\quad 64 > 16q \quad\Rightarrow\quad q < 4 \]\(\boxed{q < 4}\)
5 Find \(m\) for real and equal roots
\(mx^2 - 8x + 1 = 0\)
\(a=m, b=-8, c=1\)
\[ \text{Disc.} = 64 - 4(m)(1) = 64 - 4m \]For equal roots: Disc. \(=0\)
\[ 64 - 4m = 0 \quad\Rightarrow\quad 4m = 64 \quad\Rightarrow\quad m = 16 \]\(\boxed{m = 16}\)
📊 Key Concepts – Nature of Roots
- Discriminant: \(\Delta = b^{2} - 4ac\)
- \(\Delta > 0\) and perfect square → rational, unequal
- \(\Delta > 0\) and not perfect square → irrational, unequal
- \(\Delta = 0\) → real and equal (rational)
- \(\Delta < 0\) → imaginary (complex) and unequal