Question

How do you factor `-4.9t^{2}+19.6t+1=0`?

Answer 1

It does not factor. The quadratic formula gives us:

`t ~~ 4.05` and `t ~~ -0.05`

Explanation:

The quadratic does not factor into integer roots.

I recommend that you use the quadratic formula:

`t = (-b +-sqrt(b^2 - 4(a)(c)))/(2a)`

where `a = -4.9, b = 19.6, and c = 1`

`t ~~ 4.05` and `t ~~ -0.05`

I suspect that you will want to discard the negative root.

Answer 2

Factors of `-4.9t^2+19.6t+1=0` are

`-4.9(x-2+sqrt403.76/9.8)(x-2-sqrt403.76/9.8)=0`

Explanation:

The quadratic equation `ax^2+bx+c=0` has a solution given by

`x=(-b+-sqrt(b^2-4ac))/(2a)`.

If `alpha` and `beta` are two such roots, then

`ax^2+bx+c=0` can be factorized as `a(x-alpha)(x-beta)=0`

As can be seen nature of roots critically depends on the determinant `b^2-4ac`. The only difference being that we have a variable `t` instead of `x` and equation is `-4.9t^2+19.6t+1=0`

Here we have `a=-4.9`, `b=19.6` and `c=1`

and determinant is `19.6^2-4xx(-4.9)xx1=384.16+19.6=403.76` and roots are

`(-19.6+sqrt403.76)/(-9.8)` and `(-19.6-sqrt403.76)/(-9.8)`

and factors of `-4.9t^2+19.6t+1=0` are

`-4.9(x-(-19.6+sqrt403.76)/(-9.8))(x-(-19.6-sqrt403.76)/(-9.8))=0`

or `-4.9(x-2+sqrt403.76/9.8)(x-2-sqrt403.76/9.8)=0`

Answer 3

`0 = -4.9t^2+19.6t+1 = -49/10(t-2-sqrt(206)/7)(t-2+sqrt(206)/7)`

Explanation:

Given:

`-4.9t^2+19.6t+1 = 0`

Note that `49 = 7^2` and `196 = 4*49`, so we can break down the equation like this:

`0 = -4.9t^2+19.6t+1`

`color(white)(0) = -1/10(49t^2-196t-10)`

`color(white)(0) = -1/10((7t)^2-2(14)(7t)+(14)^2-(14)^2-10)`

`color(white)(0) = -1/10((7t-14)^2-206)`

`color(white)(0) = -1/10((7t-14)^2-(sqrt(206))^2)`

`color(white)(0) = -1/10((7t-14)-sqrt(206))((7t-14)+sqrt(206))`

`color(white)(0) = -1/10(7t-14-sqrt(206))(7t-14+sqrt(206))`

`color(white)(0) = -49/10(t-2-sqrt(206)/7)(t-2+sqrt(206)/7)`