Solve using the quadratic formula. $\newline$ $3p^2 + 2p - 6 = 0$ $\newline$ Write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth. $\newline$ $p =$ _ or $p =$ _

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Algebra 2

Solve a quadratic equation using the quadratic formula

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Q. Solve using the quadratic formula.

\newline

3p^2 + 2p - 6 = 0

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

\newline

p =

_____ or

p =

_____

Identify coefficients: Identify the coefficients of the quadratic equation. $\newline$ The quadratic equation is in the form $ap^2 + bp + c = 0$ . For the equation $3p^2 + 2p - 6 = 0$ , the coefficients are: $\newline$ a = $3$ $\newline$ b = $2$ $\newline$ c = $-6$
Substitute into formula: Substitute the coefficients into the quadratic formula. $\newline$ The quadratic formula is $p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substituting the values of $a$ , $b$ , and $c$ , we get: $\newline$ $p = \frac{-(2) \pm \sqrt{(2)^2 - 4\cdot3\cdot(-6)}}{2\cdot3}$
Simplify discriminant: Simplify the expression under the square root (the discriminant). $\newline$ Calculate the discriminant $\sqrt{(2)^2 - 4\cdot 3\cdot (-6)}$ : $\newline$ $\sqrt{4 + 72}$ $\newline$ $\sqrt{76}$
Simplify quadratic formula: Simplify the quadratic formula with the calculated discriminant. $\newline$ Now we have: $\newline$ $p = \frac{{-2 \pm \sqrt{76}}}{{6}}$
Calculate possible solutions: Calculate the two possible solutions for $p$ . $\newline$ First solution: $\newline$ $p = \frac{{-2 + \sqrt{76}}}{{6}}$ $\newline$ Second solution: $\newline$ $p = \frac{{-2 - \sqrt{76}}}{{6}}$
Simplify and round solutions: Simplify the solutions and, if necessary, round to the nearest hundredth. $\newline$ First solution: $\newline$ $p = \frac{-2 + \sqrt{76}}{6}$ $\newline$ $p = \frac{-2 + 8.72}{6}$ $\newline$ $p = \frac{6.72}{6}$ $\newline$ $p = 1.12$ $\newline$ Second solution: $\newline$ $p = \frac{-2 - \sqrt{76}}{6}$ $\newline$ $p = \frac{-2 - 8.72}{6}$ $\newline$ $p = \frac{-10.72}{6}$ $\newline$ $p = -1.79$