Question

if a² - 3a-1 = 0, find the value of a^2+1/a^2
answer with steps please​

Answer 1

Qᴜᴇsᴛɪᴏɴ :-

if a² - 3a-1 = 0, find the value of a^2+1/a^2 ?

Sᴏʟᴜᴛɪᴏɴ :-

→ a² - 3a - 1 = 0

Taking a common ,

→ a(a - 3 - 1/a) = 0

→ (a - 1/a) = 3

Squaring Both sides now,

→ (a - 1/a)² = (3)²

using (x - y)² = x² + y² - 2xy in LHS we get,

→ (a)² + (1/a)² - 2 * (a) * (1/a) = 9

→ a² + 1/a² - 2 = 9

→ (a² + 1/a²) = 9 + 2

→ (a² + 1/a²) = 11 (Ans.)

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Extra :-

• (A+B)² = A² + 2•A•B + B²

• (A-B)² = A² - 2•A•B + B²

• A² - B² = (A+B)•(A-B)

• (A+B)² = (A-B)² + 4•A•B

• (A-B)² = (A+B)² - 4•A•B

• (A+B)³ = A³ + 3•A•B•(A+B) + B³

• (A-B)³ = A³ - 3•A•B•(A-B) + B³

• A³ + B³ = (A+B)(A² - A•B + B²)

• A³ - B³ = (A-B)(A² + A•B + B²)

• (A+B+C)² = A² + B² + C² + 2•(A•B + B•C + C•A)

Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

${\star{\sf{ \: {a}^{2} - 3a - 1 }}}$

${\bf{\blue{\underline{Now:}}}}$

Divide by a,

${\implies{\sf{ \: \frac{ {a}^{2} }{a} - \frac{3a}{a} - \frac{1}{a} = 0}}}$

${\implies{\sf{ \: a - 3 - \frac{1}{a} = 0}}}$

${\implies{ \boxed{\sf{ \: a - \frac{1}{a} = 3}}}}$

Squaring both side,

${\implies{\sf{ \: \bigg( a - \frac{1}{a} \bigg)^{2} = ( {3)}^{2} }}}$

${\implies{ \boxed{\sf{\purple { \:(x - y) ^{2} } = {x}^{2} + {y}^{2} - 2xy }}}}$

${\implies{\sf{ \: {a}^{2} + \frac{1}{ {a}^{2} } - 2 \times a \times \frac{1}{a} = 9}}}$

${\implies{\sf{ \: {a}^{2} + \frac{1}{ {a}^{2} } - 2 = 9}}}$

${\implies{\sf{ \: {a}^{2} + \frac{1}{ {a}^{2} } = 9 + 2}}}$

${ \boxed{\purple{\sf{ \: {a}^{2} + \frac{1}{ {a}^{2} } = 11}}}}$