Solve Using the Quadratic Formula 2p^2=-6p+1

$2 p^{2} = - 6 p + 1$

Step 1

Move all the expressions to the left side of the equation.

Add $6 p$ to both sides of the equation.

$2 p^{2} + 6 p = 1$

Subtract $1$ from both sides of the equation.

$2 p^{2} + 6 p - 1 = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 2$ , $b = 6$ , and $c = - 1$ into the quadratic formula and solve for $p$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (2 \cdot - 1)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$p = \frac{- 6 \pm \sqrt{36 - 4 \cdot 2 \cdot - 1}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 1$ .

Multiply $- 4$ by $2$ .

$p = \frac{- 6 \pm \sqrt{36 - 8 \cdot - 1}}{2 \cdot 2}$

Multiply $- 8$ by $- 1$ .

$p = \frac{- 6 \pm \sqrt{36 + 8}}{2 \cdot 2}$

Add $36$ and $8$ .

$p = \frac{- 6 \pm \sqrt{44}}{2 \cdot 2}$

Rewrite $44$ as $2^{2} \cdot 11$ .

Factor $4$ out of $44$ .

$p = \frac{- 6 \pm \sqrt{4 (11)}}{2 \cdot 2}$

Rewrite $4$ as $2^{2}$ .

$p = \frac{- 6 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 2}$

Pull terms out from under the radical.

$p = \frac{- 6 \pm 2 \sqrt{11}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$p = \frac{- 6 \pm 2 \sqrt{11}}{4}$

Simplify $\frac{- 6 \pm 2 \sqrt{11}}{4}$ .

$p = \frac{- 3 \pm \sqrt{11}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$p = \frac{- 3 \pm \sqrt{11}}{2}$

Decimal Form:

$p = 0.15831239 \dots, - 3.15831239 \dots$