Question

How do you solve `3x^2 - 14x - 5 = 0`?

Answer 1

`x_(1,2) = (14 +- 16)/6`

Explanation:

You could solve this quadratic equation by using the quadratic formula.

For a quadratic equation that takes the general form

`color(blue)(ax^2 + bx + c = 0)`

the two solutions can be determined by using the quadratic formula

`color(blue)(x_(1,2) = (-b +- sqrt(b^2 - 4ac))/(2a)`

In your case, `a=3`, `b=-14`, and `c=-5`. This means that the two solutions will be

`x_(1,2) = (-(-14) +- sqrt( (-14)^2 - 4 * 3 * (-5)))/(2 * 3)`

`x_(1,2) = (14 +- sqrt(256))/6`

`x_(1,2) = (14 +- 16)/6 = {(x_1 = (14 + 16)/6 = color(green)(5)), (x_2 = (14 - 16)/6 = color(green)(-1/3)) :}`

Answer 2

Solve y = 3x^2 - 14x - 5 = 0 (1)

Ans: (- 1/3) and (5).

Explanation:

I use the new Transforming Method.
Transformed y' = x^2 - 14x - 15 = 0. (2)
Since (a -b + c = 0) --> 2 real roots of (2) are (-1) and (-c/a = 15).
Back to equation (1), the 2 real roots are: `(- 1/3)` and `(15/3 = 5)`