# How do you use the quadratic formula to solve #16t^2-4t+3=0#?

Given:

This is a quadratic equation in standard form:

Its roots can be found using the quadratic formula:

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To use the quadratic formula to solve (16t^2 - 4t + 3 = 0), follow these steps:

- Identify the coefficients (a), (b), and (c) in the quadratic equation (ax^2 + bx + c = 0).
- Substitute the values of (a), (b), and (c) into the quadratic formula: (x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}).
- Calculate the discriminant, (b^2 - 4ac), to determine the nature of the roots.
- If the discriminant is positive, there are two distinct real roots. If it's zero, there is one real root (a repeated root). If it's negative, there are two complex roots.
- Substitute the values of (a), (b), and (c) into the quadratic formula and simplify to find the values of (t).

For the equation (16t^2 - 4t + 3 = 0):

- (a = 16)
- (b = -4)
- (c = 3)

Substitute these values into the quadratic formula: [ t = \frac{{-(-4) \pm \sqrt{{(-4)^2 - 4(16)(3)}}}}{{2(16)}} ]

Calculate the discriminant: [ b^2 - 4ac = (-4)^2 - 4(16)(3) = 16 - 192 = -176 ]

Since the discriminant ((-176)) is negative, there are two complex roots.

Substitute the values of (a), (b), and (c) into the quadratic formula and simplify: [ t = \frac{{4 \pm \sqrt{{-176}}}}{{32}} ] [ t = \frac{{4 \pm \sqrt{{176i}}}}{{32}} ] [ t = \frac{{4 \pm 4\sqrt{{11}}i}}{{32}} ] [ t = \frac{{1 \pm \sqrt{{11}}i}}{{8}} ]

Therefore, the solutions to the equation (16t^2 - 4t + 3 = 0) are (t = \frac{{1 + \sqrt{{11}}i}}{{8}}) and (t = \frac{{1 - \sqrt{{11}}i}}{{8}}), where (i) is the imaginary unit.

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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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