Hence find the exact solutions of the equation x²–8x+11 = 1
Hence find the exact solutions of the equation x²–8x+11 = 1
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Hence find the exact solutions of the equation x²–8x+11 = 1
To solve the equation ( x^2 – 8x + 11 = 1 ), first, we need to rearrange it into standard form.
1. Subtract 1 from both sides:
[
x^2 – 8x + 10 = 0
]
2. Now we can factor this quadratic equation. We look for two numbers that multiply to ( 10 ) (the constant term) and add up to ( -8 ) (the coefficient of ( x )). The numbers ( -5 ) and ( -2 ) fit this requirement.
3. Thus, we can factor the equation as:
[
(x – 5)(x – 2) = 0
]
4. Setting each factor to zero gives us the solutions:
[
x – 5 = 0 quad Rightarrow quad x = 5
]
[
x – 2 = 0 quad Rightarrow quad x = 2
]
Therefore, the exact solutions to the equation ( x^2 – 8x + 11 = 1 ) are ( x = 5 ) and ( x = 2 ).