Use the quadratic formula to solve. Express your answer in simplest form.
9, p, squared, plus, 17, p, plus, 12, equals, 3, p, squared
9p
2
+17p+12=
3p
2
To solve the equation using the quadratic formula, we need to rewrite the equation in the form of "ax^2 + bx + c = 0".
The given equation is: 9p^2 + 17p + 12 = 3p^2
Move all the terms to one side of the equation to obtain: 9p^2 - 3p^2 + 17p + 12 = 0
Combining like terms, we have: 6p^2 + 17p + 12 = 0
Now we can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our given equation, a = 6, b = 17, and c = 12.
Solving for p using the quadratic formula, we have:
p = (-17 ± √(17^2 - 4(6)(12))) / (2(6))
Simplifying the expression under the square root:
p = (-17 ± √(289 - 288)) / 12
p = (-17 ± √1) / 12
Since the square root of 1 is 1, we have:
p = (-17 ± 1) / 12
Simplifying further:
p1 = (-17 + 1) / 12 = -16 / 12 = -4/3
p2 = (-17 - 1) / 12 = -18 / 12 = -3/2
Therefore, the solutions to the equation are p = -4/3 and p = -3/2.