How do you find the roots, real and imaginary, of #y= x^2 - 12x-1 # using the quadratic formula?
See a solution process below:
Substituting:
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The solutions are
The quadratic equation is
Our equation is
Start by calculating the discriminant
As,
The roots are
So,
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This quadratic has zeros
Note that:
is in the standard form
This has zeros given by the quadratic formula:
Bonus
Define:
Note that the recurrence rule is chosen such that the sequence:
The first few terms of this sequence are:
Then:
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To find the roots of the quadratic equation y = x^2 - 12x - 1 using the quadratic formula:
-
Identify the coefficients a, b, and c in the quadratic equation y = ax^2 + bx + c. In this case, a = 1, b = -12, and c = -1.
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Substitute the values of a, b, and c into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
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Plug in the values:
x = (-( -12) ± √((-12)^2 - 4(1)(-1))) / (2(1))
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Simplify:
x = (12 ± √(144 + 4)) / 2 = (12 ± √148) / 2 = (12 ± 2√37) / 2
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Divide both terms by 2:
x = (12/2 ± 2√37/2) = (6 ± √37)
So, the roots of the equation y = x^2 - 12x - 1 are x = 6 + √37 and x = 6 - √37. These roots can be considered as real and irrational.
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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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