Question

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~ Solve the equation by Using the formulae x^2 - 6x = 27 ​

Answer 1

$\sf{\underline{\boxed{\green{\large{\bold{ Question}}}}}}$

• solve the equation using formulae $\sf x^2 - 6x = 27$

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$\sf{\underline{\boxed{\green{\large{\bold{ Solution}}}}}}$

$\sf\implies x^2 - 6x = 27$

$\sf\implies x^2 - 6x - 27 = 0$

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• compare the eq with $\sf{\underline{\bold{ax^2 + bx + c = 0 }}}$

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☯ a = 1

☯ b = -6

☯ c = -27

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now :-

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$\sf{\underline{\boxed{\pink{\large{\mathfrak{x = \dfrac{ - b \pm \sqrt D }{2a }}}}}}}$

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$\sf{\underline{\boxed{\pink{\large{\mathfrak{ D = b^2 - 4ac }}}}}}$

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• finding value of D.

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$\sf\implies D = b^2 - 4ac$

$\sf\implies D = (-6)^2 - 4 \times 1 \times -27$

$\sf\implies D = 36 + 108$

$\sf\implies D = 144$

$\sf{\underline{\boxed{\blue{\large{\bold{ D = 144}}}}}}$

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• putting values in the eq.

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$\sf\implies x = \dfrac{ -b \pm\sqrt D }{2a}$

$\sf\implies x = \dfrac{ -( -6) \pm\sqrt {144} }{2\times 1 }$

$\sf\implies x = \dfrac{ 6 \pm 12 }{2}$

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✒ $\sf x = \dfrac{ 6 + 12 }{ 2 }$

$\implies x = \dfrac {18}{2}$

$\implies x = 9$

$\sf{\underline{\boxed{\purple{\large{\bold{ x = 9 }}}}}}$

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✒ $\sf x = \dfrac{ 6 - 12 }{ 2 }$

$\implies x = \dfrac {-6}{2}$

$\implies x = -3$

$\sf{\underline{\boxed{\purple{\large{\bold{ x = -3 }}}}}}$

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$\sf{\underline{\boxed{\purple{\large{\bold{ x = 9 \: or \:-3 }}}}}}$

Answer 2

Solution :

$\bigstar$ By using quadratic formula :

We have equation p(x) = x² - 6x = 27

∴ x² - 6x - 27 = 0

As we know that given quadratic polynomial compared with ax² + bx + c;

• a = 1
• b = -6
• c = -27

$\boxed{\bf{x=\frac{-b\pm\sqrt{b^{2}-4ac} }{2a} }}}$

$\longrightarrow\tt{x=\dfrac{-(-6)\pm\sqrt{(-6)^{2}-4\times 1\times (-27)} }{2\times 1} }\\\\\\\longrightarrow\tt{x=\dfrac{6\pm\sqrt{36-4\times (-27)} }{2} }\\\\\\\longrightarrow\tt{x=\dfrac{6\pm\sqrt{36+108} }{2} }\\\\\\\longrightarrow\tt{x=\dfrac{6\pm\sqrt{144} }{2} }\\\\\\\longrightarrow\tt{x=\dfrac{6\pm 12}{2}} \\\\\\\longrightarrow\tt{x=\dfrac{6+12}{2} \:\:\:Or\:\:\:x=\dfrac{6-12}{2} }\\\\\\\longrightarrow\tt{x=\cancel{\dfrac{18}{2}} \:\:\:Or\:\:\:x=\cancel{\dfrac{-6}{2} }}$

$\longrightarrow\bf{x=9\:\:\:Or\:\:\:x=-3}$

Thus;

∴ x = 9 Or x = -3 are two roots of the given polynomial .