Find the roots of the quadratic equations x^2 - 1/12x -1/12 = 0 ?
you can use the quadratic formula, and mess with all those fractions, or you can see that you really have
1/12 (12x^2-x-1) = 1/12 (4x+1)(3x-1)
Now the roots become clear.
To find the roots of a quadratic equation, we can use the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation x^2 - (1/12)x - (1/12) = 0, we can identify that:
a = 1
b = -1/12
c = -1/12
By plugging these values into the quadratic formula, we can find the roots.
Let's calculate the discriminant (b^2 - 4ac) first:
Discriminant = (-1/12)^2 - 4(1)(-1/12)(-1/12)
= 1/144 + 1/144
= 2/144
= 1/72
Now, substituting the values in the quadratic formula:
x = (-(-1/12) ± √(1/72)) / (2(1))
x = (1/12 ± √(1/72)) / 2
x = (1 ± √(1/72)) / 24
Simplifying further:
x = (1 ± √(1/72)) / 24
x = (1 ± √(1)/√(72)) / 24
x = (1 ± 1/√(72)) / 24
x = (1 ± 1/(√(9)√(8))) / 24
x = (1 ± 1/(3√(8))) / 24
x = (1 ± 1/(3 ∙ 2√(2))) / 24
Hence, the roots of the quadratic equation x^2 - (1/12)x - (1/12) = 0 are:
x = (1 + 1/(3 ∙ 2√(2))) / 24
x = (1 - 1/(3 ∙ 2√(2))) / 24