Basic Math Examples

$2 p^{2} = 6 p - 20$

Step 1

Subtract $6 p$ from both sides of the equation.

$2 p^{2} - 6 p = - 20$

Step 2

Add $20$ to both sides of the equation.

$2 p^{2} - 6 p + 20 = 0$

Step 3

Factor $2$ out of $2 p^{2} - 6 p + 20$ .

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Step 3.1

Factor $2$ out of $2 p^{2}$ .

$2 (p^{2}) - 6 p + 20 = 0$

Step 3.2

Factor $2$ out of $- 6 p$ .

$2 (p^{2}) + 2 (- 3 p) + 20 = 0$

Step 3.3

Factor $2$ out of $20$ .

$2 p^{2} + 2 (- 3 p) + 2 \cdot 10 = 0$

Step 3.4

Factor $2$ out of $2 p^{2} + 2 (- 3 p)$ .

$2 (p^{2} - 3 p) + 2 \cdot 10 = 0$

Step 3.5

Factor $2$ out of $2 (p^{2} - 3 p) + 2 \cdot 10$ .

$2 (p^{2} - 3 p + 10) = 0$

Step 4

Divide each term in $2 (p^{2} - 3 p + 10) = 0$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 (p^{2} - 3 p + 10) = 0$ by $2$ .

$\frac{2 (p^{2} - 3 p + 10)}{2} = \frac{0}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 (p^{2} - 3 p + 10)}{2} = \frac{0}{2}$

Step 4.2.1.2

Divide $p^{2} - 3 p + 10$ by $1$ .

$p^{2} - 3 p + 10 = \frac{0}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $2$ .

$p^{2} - 3 p + 10 = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 1$ , $b = - 3$ , and $c = 10$ into the quadratic formula and solve for $p$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot 10)}}{2 \cdot 1}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 3$ to the power of $2$ .

$p = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 10}}{2 \cdot 1}$

Step 7.1.2

Multiply $- 4 \cdot 1 \cdot 10$ .

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Step 7.1.2.1

Multiply $- 4$ by $1$ .

$p = \frac{3 \pm \sqrt{9 - 4 \cdot 10}}{2 \cdot 1}$

Step 7.1.2.2

Multiply $- 4$ by $10$ .

$p = \frac{3 \pm \sqrt{9 - 40}}{2 \cdot 1}$

Step 7.1.3

Subtract $40$ from $9$ .

$p = \frac{3 \pm \sqrt{- 31}}{2 \cdot 1}$

Step 7.1.4

Rewrite $- 31$ as $- 1 (31)$ .

$p = \frac{3 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 1}$

Step 7.1.5

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$p = \frac{3 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 1}$

Step 7.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$p = \frac{3 \pm i \sqrt{31}}{2 \cdot 1}$

Step 7.2

Multiply $2$ by $1$ .

$p = \frac{3 \pm i \sqrt{31}}{2}$

Step 8

The final answer is the combination of both solutions.

$p = \frac{3 + i \sqrt{31}}{2}, \frac{3 - i \sqrt{31}}{2}$