# How do you find the roots, real and imaginary, of #y=-x^2 +32x -16# using the quadratic formula?

There are no imaginary roots.

Quadratic formula:

Simplifying:

There are no imaginary roots.

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Roots are

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To find the roots of the quadratic equation (y = -x^2 + 32x - 16), you can use the quadratic formula: (x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}), where (a = -1), (b = 32), and (c = -16). Substituting these values into the formula:

(x = \frac{{-32 \pm \sqrt{{32^2 - 4(-1)(-16)}}}}{{2(-1)}})

Simplify under the square root:

(x = \frac{{-32 \pm \sqrt{{1024 - 64}}}}{{-2}})

(x = \frac{{-32 \pm \sqrt{{960}}}}{{-2}})

(x = \frac{{-32 \pm 4\sqrt{{60}}}}{{-2}})

Now, you have:

(x_1 = \frac{{-32 + 4\sqrt{{60}}}}{-2})

(x_2 = \frac{{-32 - 4\sqrt{{60}}}}{-2})

These are the roots of the quadratic equation.

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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.

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