Question

Solve itne quadratic equation:

x^2 -90x + 900=0

Answer 1

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Answer 2

Hey Nancy,

Here is your solution.



Let , α and ß are its zeroes.

In this quadratic equation,

Coefficient of x² ( a ) = 1

Coefficient of x ( b ) = -90

Constant term ( c ) = 900

Now,

⇒Sum of zeroes = -b/a

⇒ α + ß = - ( -90 ) ÷ 1

∴ α + ß = 90         --------------- ( 1 )

And,

⇒ Product of zeroes = c/a

⇒ αß = 90/1

 ∴ αß = 900   ----------- ( 2 )

To find the value of α and ß , we have to first find the value of ( α - ß ), to do so we have ( α + ß ) and ( αß ).

Let's find it !


We know that,

⇒ ( α - ß )² = α² + ß² - 2αß

⇒ ( α - ß )² = α² + ß² + 2αß - 4αß

⇒ ( α - ß )² = ( α + ß )² - 4αß

By substituting the values of ( 1 ) and ( 2 ),

⇒ ( α - ß )² = ( 90 )² - 4 × 90

⇒ ( α - ß )² = 8100 - 3600

⇒ ( α - ß )² = 4500

⇒ ( α - ß ) = √ ( 4500 )

⇒ ( α - ß ) = √ ( 2 × 2 × 3 × 3 × 5 × 5 × 5 )

⇒ ( α - ß ) = √ ( 2² × 3² × 5² × 5 )

⇒ ( α - ß ) = 2 × 3 × 5 √5

∴ ( α - ß ) = 30√5     ------------------- ( 3 )

By adding the ( 1 ) and ( 3 ),

⇒ α + ß + α - ß = 90 + 30√5

⇒ 2α = 2 ( 45 + 15√5 )

⇒ α = 2 ( 45 + 15√5 ) ÷ 2

∴  α = ( 45 + 15√5 )

By substituting the value of α in ( 1 ),

⇒ α + ß = 90

⇒ 45 + 15√5 + ß = 90

⇒ ß = 90 - 45 - 15√5

∴ ß = 45 - 15√5.

Now,

If α and ß be the zeroes of x² - 90x + 900.

So,

⇒ x² - 90x + 900 = ( x - α ) ( x - ß )

By substituting the value of α and ß,

⇒ x² - 90x + 900 = { x - ( 45 + 15√5 ) } { x - ( 45 - 15√5 ) }

⇒ x² - 90x + 900 = ( x - 45 - 15√5 ) ( x - 45 + 15√5 )



The required answer is ( x - 45 - 15√5 ) ( x - 45 + 15√5 ).



Hope it helps !!