# How do you find the roots, real and imaginary, of #y=-x^2+5x-9 # using the quadratic formula?

real roots: none

imaginary roots:

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To find the roots of ( y = -x^2 + 5x - 9 ) using the quadratic formula, we identify the coefficients: ( a = -1 ), ( b = 5 ), and ( c = -9 ).

Substitute these values into the quadratic formula:

[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} ]

[ x = \frac{{-5 \pm \sqrt{{5^2 - 4(-1)(-9)}}}}{{2(-1)}} ]

[ x = \frac{{-5 \pm \sqrt{{25 - 36}}}}{{-2}} ]

[ x = \frac{{-5 \pm \sqrt{{-11}}}}{{-2}} ]

The discriminant ( b^2 - 4ac ) is negative, indicating that the roots will be imaginary.

So, the roots are:

[ x = \frac{{-5}}{{-2}} + \frac{{i \sqrt{11}}}{{-2}} ] [ x = \frac{5}{2} - \frac{{i \sqrt{11}}}{2} ]

[ x = \frac{{-5}}{{-2}} - \frac{{i \sqrt{11}}}{{-2}} ] [ x = -\frac{5}{2} + \frac{{i \sqrt{11}}}{2} ]

Therefore, the roots of the equation are ( x = \frac{5}{2} - \frac{{i \sqrt{11}}}{2} ) and ( x = -\frac{5}{2} + \frac{{i \sqrt{11}}}{2} ).

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To find the roots of the quadratic equation (y = -x^2 + 5x - 9), we can use the quadratic formula, which is given by:

[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}]

where (a), (b), and (c) are the coefficients of the quadratic equation (ax^2 + bx + c = 0).

For the given equation (y = -x^2 + 5x - 9), we have (a = -1), (b = 5), and (c = -9).

Plugging these values into the quadratic formula:

[x = \frac{{-5 \pm \sqrt{{5^2 - 4(-1)(-9)}}}}{{2(-1)}}]

[x = \frac{{-5 \pm \sqrt{{25 - 36}}}}{{-2}}]

[x = \frac{{-5 \pm \sqrt{{-11}}}}{{-2}}]

Since the discriminant ((b^2 - 4ac)) is negative, the roots will be complex. Thus, the roots are:

[x = \frac{{-5 + \sqrt{{-11}}}}{{-2}}] and [x = \frac{{-5 - \sqrt{{-11}}}}{{-2}}]

Therefore, the roots are complex and cannot be expressed as real numbers.

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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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