(a) Find the value of m for which the quadratic equation (m - 1)x2 -

1.	(a) Find the value of m for which the quadratic equation (m - 1)x2 - 2 (m - 1) x +1 - 0 has two real and equal roots.(b) Solve the following quadratic equation for x:√3x2 + 10x + 7√3 = 0
Answer» (a) D = 0 ⇒ b² - 4ac = 0 ⇒ 4(m - 1)² - 4(m - 1) = 0 ⇒ 4(m - 1) (m - 1 - 1) = 0 ⇒ (m - 1) (m - 2) = 0 ⇒ m = 1 or m = 2 \(\because\) Given equation (m - 1)x² + 2(m - 1)x + 1 = 0 is a quadratic equation. \(\therefore\) m \(\neq\)1(If m = 1 then equation (1) is not remaining a quadratic equation) \(\therefore\) m = 2 (b) \(\sqrt3\) x² + 10x + 7\(\sqrt3\) = 0 ⇒ \(\sqrt3\) x² + 3x + 7x + 7\(\sqrt3\) = 0 ⇒ \(\sqrt3\) x (x + \(\sqrt3\)) + 7(x + \(\sqrt3\)) = 0 ⇒ (x + \(\sqrt3\)) (\(\sqrt3\)x + 7) = 0 ⇒ x + \(\sqrt3\) = 0 or \(\sqrt3\)x + 7 = 0 ⇒ x = -\(\sqrt3\) or x = -7/\(\sqrt3\)

(a) Find the value of m for which the quadratic equation (m - 1)x2 - 2 (m - 1) x +1 - 0 has two real and equal roots.(b) Solve the following quadratic equation for x:√3x2 + 10x + 7√3 = 0

Answer»

(a) D = 0

⇒ b² - 4ac = 0

⇒ 4(m - 1)² - 4(m - 1) = 0

⇒ 4(m - 1) (m - 1 - 1) = 0

⇒ (m - 1) (m - 2) = 0

⇒ m = 1 or m = 2

\(\because\) Given equation (m - 1)x² + 2(m - 1)x + 1 = 0 is a quadratic equation.

\(\therefore\) m \(\neq\)1(If m = 1 then equation (1) is not remaining a quadratic equation)

\(\therefore\) m = 2

(b) \(\sqrt3\) x² + 10x + 7\(\sqrt3\) = 0

⇒ \(\sqrt3\) x² + 3x + 7x + 7\(\sqrt3\) = 0

⇒ \(\sqrt3\) x (x + \(\sqrt3\)) + 7(x + \(\sqrt3\)) = 0

⇒ (x + \(\sqrt3\)) (\(\sqrt3\)x + 7) = 0

⇒ x + \(\sqrt3\) = 0 or \(\sqrt3\)x + 7 = 0

⇒ x = -\(\sqrt3\) or x = -7/\(\sqrt3\)

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