Question

plzzz correct answer chapter quadratic equation for board exam​

Answer 1

Question:

Solve the following quadratic equations:

i) x² - 11x = 42

$\displaystyle{\sf\:ii\:)\:x\:+\:\dfrac{1}{x}\:=\:3\:\dfrac{1}{3}\:\qquad\cdots\:x\:\neq\:0}$

Answer:

$\displaystyle{\sf\:i\:)\:\boxed{\pink{\sf\:x\:=\:14\:}}\sf\:\quad\:OR\:\quad\:\boxed{\pink{\sf\:x\:=\:-\:3\:}}}$

$\displaystyle{\sf\:ii\:)\:\boxed{\red{\sf\:x\:=\:3\:}}\sf\:\quad\:OR\:\quad\:\boxed{\red{\sf\:x\:=\:\dfrac{1}{3}}}}$

Step-by-step-explanation:

i)

The given quadratic equation is

x² - 11x = 42.

⇒ x² - 11x - 42 = 0

⇒ x² - 14x + 3x - 42 = 0

⇒ x ( x - 14 ) + 3 ( x - 14 ) = 0

⇒ ( x - 14 ) ( x + 3 ) = 0

⇒ ( x - 14 ) = 0 $\displaystyle{\qquad\sf\:OR\:\qquad}$ ( x + 3 ) = 0

⇒ x - 14 = 0 $\displaystyle{\qquad\sf\:OR\:\qquad}$ x + 3 = 0

$\displaystyle{\therefore\:\underline{\boxed{\pink{\sf\:x\:=\:14\:}}}\sf\:\quad\:OR\:\quad\:\underline{\boxed{\pink{\sf\:x\:=\:-\:3\:}}}}$

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ii)

The given quadratic equation is

$\displaystyle{\sf\:x\:+\:\dfrac{1}{x}\:=\:3\:\dfrac{1}{3}}$

$\displaystyle{\implies\sf\:\dfrac{x^2\:+\:1}{x}\:=\:\dfrac{3\:\times\:3\:+\:1}{3}}$

$\displaystyle{\implies\sf\:\dfrac{x^2\:+\:1}{x}\:=\:\dfrac{9\:+\:1}{3}}$

$\displaystyle{\implies\sf\:\dfrac{x^2\:+\:1}{x}\:=\:\dfrac{10}{3}}$

$\displaystyle{\implies\sf\:3\:(\:x^2\:+\:1\:)\:=\:10\:x}$

$\displaystyle{\implies\sf\:3x^2\:+\:3\:=\:10x}$

$\displaystyle{\implies\sf\:3x^2\:-\:10x\:+\:3\:=\:0}$

$\displaystyle{\implies\sf\:3x^2\:-\:9x\:-\:x\:+\:3\:=\:0}$

$\displaystyle{\implies\sf\:3x\:(\:x\:-\:3\:)\:-\:1\:(\:x\:-\:3\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:x\:-\:3\:)\:(\:3x\:-\:1\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:x\:-\:3\:)\:=\:0\:\quad\:OR\:\quad\:(\:3x\:-\:1\:)\:=\:0}$

$\displaystyle{\implies\sf\:x\:-\:3\:=\:0\:\quad\:OR\:\quad\:3x\:-\:1\:=\:0}$

$\displaystyle{\implies\sf\:x\:=\:3\:\quad\:OR\:\quad\:3x\:=\:1}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:x\:=\:3\:}}}\sf\:\quad\:OR\:\quad\:\underline{\boxed{\red{\sf\:x\:=\:\dfrac{1}{3}}}}}$