If 𝛼α and 𝛽β are the roots of the equation 𝑥2−5𝑥+𝑐=0x 2 −5x+c=0, and 𝛼−𝛽=1α−β=1 then 𝑐=?c=?.
Question
If 𝛼α and 𝛽β are the roots of the equation 𝑥2−5𝑥+𝑐=0x 2 −5x+c=0, and 𝛼−𝛽=1α−β=1 then 𝑐=?c=?.
Solution
The roots of a quadratic equation ax² + bx + c = 0 are given by the formula:
α, β = [-b ± sqrt(b² - 4ac)] / 2a
Given that α - β = 1, we can substitute the values of α and β from the formula into this equation:
1 = [-(-5) - sqrt((-5)² - 41c)] / 21 - [-(-5) + sqrt((-5)² - 41c)] / 21 1 = [5 - sqrt(25 - 4c)] / 2 - [5 + sqrt(25 - 4c)] / 2 1 = - sqrt(25 - 4c) / 2
Squaring both sides to get rid of the square root:
1 = 25 - 4c 4c = 25 - 1 4c = 24 c = 24 / 4 c = 6
So, the value of c that satisfies the given conditions is 6.
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