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## Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 3 Algebra Ex 3.10

Question 1.

Solve the following quadratic equations by factorization method

(i) 4x^{2} – 7x – 2 = 0

(ii) 3(p^{2} – 6) = p(p + 5)

(iii) \(\sqrt{a(a-7)}=3 \sqrt{2}\)

(iv) \(\sqrt{2} x^{2}+7 x+5 \sqrt{2}=0\)

(v) 2x^{2} – x + \(\frac{1}{8}\) = 0

Solution:

(ii) 3(p^{2} – 6) = p(p + 5)

3p^{2} – 18 = p^{2} + 5p ⇒ 39^{2} – 5p – 18 = 0

⇒ 2p^{2} – 5p – 18 = 0

⇒ (2p – 9) (p + 2) = 0 ⇒ p = \(\frac{9}{2}\), -2

(iii) \(\sqrt{a(a-7)}=3 \sqrt{2}\)

Squaring on both sides

a(a – 7) = 9 × 2

a^{2} – 7a – 18 = 0

a^{2} – 9a + 2a – 18 = 0

a(a – 9) + 2(a – 9) = 0

(a – 9) (a + 2) = 0

⇒ a = 9, a = -2

(v) 2x^{2} – x + \(\frac{1}{8}\) = 0

16x^{2} – 8x + 1 = 0

16x^{2} – 4x – 4x + 1 = 0

4x(4x – 1) – 1(4x – 1) = 0

(4x – 1) (4x – 1) = 0

⇒ x = \(\frac{1}{4}, \frac{1}{4}\)

Question 2.

The number of volleyball games that must be scheduled in a league with n teams is given by G(n) = \(\frac{n^{2}-n}{2}\) where each team plays with every other team exactly once. A league schedules 15 games. How many teams are in the league?

Answer:

Number of games = 15

G(n) = \(\frac{n^{2}-n}{2}\)

\(\frac{n^{2}-n}{2}\) = 15

n^{2} – n = 30 ⇒ n^{2} – n – 30 = 0

⇒ n^{2}– 6n – 5n – 30 = 0

(n – 6) (n + 5) = 0

n – 6 = 0 or n + 5 = 0

[Note: – 5 is neglected because number of team is not negative]

n = 6 or n = -5

∴ Number of teams = 6