# How do you find the discriminant, describe the number and type of root, and find the exact solution using the quadratic formula given #2x-5=-x^2#?

Roots are

and hence roots are irrational and conjugate.

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To find the discriminant, number and type of roots, and the exact solution using the quadratic formula for the equation (2x - 5 = -x^2):

- Begin by rewriting the equation in standard form: (x^2 + 2x - 5 = 0).
- Identify the coefficients: (a = 1), (b = 2), and (c = -5).
- Calculate the discriminant using the formula (\Delta = b^2 - 4ac).
- Substitute the values: (\Delta = (2)^2 - 4(1)(-5)).
- Simplify to find (\Delta).
- Determine the nature of the roots:
- If (\Delta > 0), the equation has two distinct real roots.
- If (\Delta = 0), the equation has one real root (a repeated root).
- If (\Delta < 0), the equation has two complex roots.

- Use the quadratic formula to find the exact solutions:
- (x = \frac{{-b \pm \sqrt{\Delta}}}{{2a}}).
- Substitute the values of (a), (b), and (\Delta).
- Simplify to find the exact solutions for (x).

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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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