Question

Solve the equation for all real solutions in simplest form.
3b^{2}-7b+3= 6

Answer 1

b=(7-sqrd85/6), b=(7+sqrd85/6)

Explanation:

3b^2-7b+3=6

3b^2-7b+3-6=0

3b^2-7b-3=0

a=3, b=-7, c=-3

b=7+-sqrd49+36/6

b=(7-sqrd85/6), b=(7+sqrd85/6)

Answer 2

`\displaystyle (7 + √(85))/(6)` and
`\displaystyle (7 - √(85))/(6)`.

Explanation:

(Replace
`b` with
`x` to avoid confusion with symbols in the quadratic equation.)

Notice that the equation
`3\, x^(2) - 7\, x + 3 = 6` is quadratic with respect to the unknown
`x`. Rewrite the equation in standard form
`a\, x^(2) + b\, x + c = 0` before applying the quadratic equation:

`3\, x^(2) - 7\, x + 3 = 6`.

`3\, x^(2) - 7\, x + 3 - 6 = 0`.

`3\, x^(2) + (- 7)\, x + (-3) = 0`.

Thus, for the quadratic equation,
`a = 3`,
`b = (-7)`, and
`c = (-3)`. Apply the quadratic equation to find the solutions:

`\begin{aligned}x &= \frac{-\, b + \sqrt{b^(2) - 4\, a\, c}}{2\, a} \\ &= \frac{-(-7) + \sqrt{(-7)^(2) - 4 * 3 * (-3)}}{2 * 3} \\ &= (7 + √(85))/(6)\end{aligned}`.

`\begin{aligned}x &= \frac{-\, b - \sqrt{b^(2) - 4\, a\, c}}{2\, a} \\ &= \frac{-(-7) - \sqrt{(-7)^(2) - 4 * 3 * (-3)}}{2 * 3} \\ &= (7 - √(85))/(6)\end{aligned}`.