Full solution

Q. Solve using the quadratic formula.

\newline

6w^2 + 7w + 2 = 0

\newline

Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.

\newline

w =

_____ or

w =

_____

Identify values: Identify the values of $a$ , $b$ , and $c$ in the quadratic equation $6w^2 + 7w + 2 = 0$ . By comparing the equation to the standard form $ax^2 + bx + c = 0$ , we find: $a = 6$ $b = 7$ $c = 2$
Substitute into formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula $w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substitute $a = 6$ , $b = 7$ , and $c = 2$ into the formula. $w = \frac{-(7) \pm \sqrt{(7)^2 - 4\cdot6\cdot2}}{2\cdot6}$
Simplify and calculate discriminant: Simplify the expression under the square root and calculate the discriminant $\sqrt{b^2 - 4ac}$ . $\sqrt{(7)^2 - 4\cdot 6\cdot 2}$ $= \sqrt{49 - 48}$ $= \sqrt{1}$
Calculate solutions: Calculate the two possible solutions for $w$ . $\newline$ Since the discriminant is $1$ , we have: $\newline$ $w = \frac{-7 \pm 1}{2 \times 6}$ $\newline$ This gives us two solutions: $\newline$ $w = \frac{-7 + 1}{12}$ and $w = \frac{-7 - 1}{12}$
Simplify solutions: Simplify both solutions. $\newline$ For the first solution: $\newline$ $w = \frac{-7 + 1}{12}$ $\newline$ $w = \frac{-6}{12}$ $\newline$ $w = -\frac{1}{2}$ $\newline$ For the second solution: $\newline$ $w = \frac{-7 - 1}{12}$ $\newline$ $w = \frac{-8}{12}$ $\newline$ $w = -\frac{2}{3}$ $\newline$ Both fractions are already in simplest form.
Simplify solutions: Simplify both solutions. $\newline$ For the first solution: $\newline$ $w = \frac{-7 + 1}{12}$ $\newline$ $w = \frac{-6}{12}$ $\newline$ $w = -\frac{1}{2}$ $\newline$ For the second solution: $\newline$ $w = \frac{-7 - 1}{12}$ $\newline$ $w = \frac{-8}{12}$ $\newline$ $w = -\frac{2}{3}$ $\newline$ Both fractions are already in simplest form.If necessary, round the solutions to the nearest hundredth. $\newline$ However, since both solutions are fractions in simplest form, there is no need to round to the nearest hundredth.